title: Aneuploidy

text: A research work published in 2008 focuses on the mechanisms involved in aneuploidy formation, specifically on the epigenetic origin of aneuploid cells. Epigenetic inheritance is defined as cellular information other than the DNA sequence itself, that is still heritable during cell division. DNA methylation and histone modifications comprise two of the main epigenetic modifications important for many physiological and pathological conditions, including cancer. Aberrant DNA methylation is the most common molecular lesion in cancer-cells, even more frequent than gene mutations. Tumor suppressor gene silencing by CpG island promoter hypermethylation is supposed to be the most frequent epigenetic modification in cancer cells. Epigenetic characteristics of cells may be modified by several factors including environmental exposure, deficiencies of certain nutrients, radiation, etc. Some of the alterations have been correlated with the formation of aneuploid cells in vivo. In this study it is suggested on a growing basis of evidence, that not only genetics but also epigenetics, contribute to aneuploid cell formation.


title: Aneuploidy

text: The terms "partial monosomy" and "partial trisomy" are used to describe an imbalance of genetic material caused by loss or gain of part of a chromosome. In particular, these terms would be used in the situation of an unbalanced translocation, where an individual carries a derivative chromosome formed through the breakage and fusion of two different chromosomes. In this situation, the individual would have three copies of part of one chromosome (two normal copies and the portion that exists on the derivative chromosome) and only one copy of part of the other chromosome involved in the derivative chromosome. Robertsonian translocations, for example, account for a very small minority of Down syndrome cases (<5%). The formation of one isochromosome results in partial trisomy of the genes present in the isochromosome and partial monosomy of the genes in the lost arm.


title: Aneuploidy

text: Agents capable of causing aneuploidy are called aneugens. Many mutagenic carcinogens are aneugens. X-rays, for example, may cause aneuploidy by fragmenting the chromosome; it may also target the spindle apparatus. Other chemicals such as colchicine can also produce aneuploidy by affecting microtubule polymerization.


title: Aneuploidy

text: Exposure of males to lifestyle, environmental and/or occupational hazards may increase the risk of spermatozoa aneuploidy. Tobacco smoke contains chemicals that cause DNA damage. Smoking also can induce aneuploidy. For instance, smoking increases chromosome 13 disomy in spermatozoa by 3-fold, and YY disomy by 2-fold.


title: Aneuploidy

text: Occupational exposure to benzene is associated with a 2.8-fold increase of XX disomy and a 2.6-fold increase of YY disomy in spermatozoa.


title: Aneuploidy

text: Pesticides are released to the environment in large quantities so that most individuals have some degree of exposure. The insecticides fenvalerate and carbaryl have been reported to increase spermatozoa aneuploidy. Occupational exposure of pesticide factory workers to fenvalerate is associated with increased spermatozoa DNA damage. Exposure to fenvalerate raised sex chromosome disomy 1.9-fold and disomy of chromosome 18 by 2.6-fold. Exposure of male workers to carbaryl increased DNA fragmentation in spermatozoa, and also increased sex chromosome disomy by 1.7-fold and chromosome 18 disomy by 2.2-fold.


title: Aneuploidy

text: Humans are exposed to perfluorinated compounds (PFCs) in many commercial products. Men contaminated with PFCs in whole blood or seminal plasma have spermatozoa with increased levels of DNA fragmentation and chromosomal aneuploidies.


title: Aneuploidy

text: Germline aneuploidy is typically detected through karyotyping, a process in which a sample of cells is fixed and stained to create the typical light and dark chromosomal banding pattern and a picture of the chromosomes is analyzed. Other techniques include fluorescence in situ hybridization (FISH), quantitative PCR of short tandem repeats, quantitative fluorescence PCR (QF-PCR), quantitative PCR dosage analysis, Quantitative Mass Spectrometry of Single Nucleotide Polymorphisms, and comparative genomic hybridization (CGH).


title: Aneuploidy

text: These tests can also be performed prenatally to detect aneuploidy in a pregnancy, through either amniocentesis or chorionic villus sampling. Pregnant women of 35 years or older are offered prenatal testing because the chance of chromosomal aneuploidy increases as the mother's age increases.


title: Aneuploidy

text: Recent advances have allowed for less invasive testing methods based on the presence of fetal genetic material in maternal blood. See Triple test and Cell-free fetal DNA.


title: Aneuploidy

text: In the strict sense, a chromosome complement having a number of chromosomes other than 46 (in humans) is considered "heteroploid" while an exact multiple of the haploid chromosome complement is considered euploid.


title: Drumline (film)

text: Drumline is a 2002 American coming-of-age teen comedy-drama film directed by Charles Stone III. The screenplay, which was inspired by the Southwest Dekalb High School Drumline, was written by Tina Gordon Chism and Shawn Schepps. The film follows a young drummer from New York, played by Nick Cannon, who enters the fictional Atlanta A&T University and bumps heads with the leader of his new school's drum section. Zoe Saldana, Leonard Roberts and Orlando Jones also co-star.


title: Drumline (film)

text: The film received generally positive reviews from critics, with most of them praising the musical bands' overall performances. It was a success at the box office, earning over $56 million in the U.S., and almost $1.2 million in foreign markets.


title: Drumline (film)

text: A direct sequel "", premiered on VH1 on October 27, 2014. Cannon reprises his role as Devon, albeit as a supporting character rather than the protagonist.


title: Drumline (film)

text: The story revolves around Devon Miles, a teen who has just graduated from high school in New York City. Upon graduating, Devon heads to Atlanta, Georgia to attend Atlanta A&T University, a historically black college that takes enormous pride in its marching band. En route to A&T, Devon befriends fellow band mates Charles, Jayson, and Ernest. Devon was personally invited to attend on a full scholarship by Dr. Lee, head of the band, for his prodigious talents. The A&T band separates itself from its competitors by requiring all members to read music, by focusing on various styles of music rather than what is currently popular on the radio, and by dedication to the teamwork emphasized "one band, one sound" concept. Preseason band camp is physically and mentally challenging, designed to push members past what they previously thought were their limits. At the end of preseason, the musicians audition for spots on the field, and Devon is the only freshman to make P1, the highest-level player. While going through this rigorous process, Devon also finds time to romance an upperclassman dancer, Laila.


title: Drumline (film)

text: College life starts well for Devon, as he has a girlfriend and a spot on the field. Things begin to sour when Sean, Devon's percussion leader, begins to grow weary of his cocky attitude. Sean later challenges Devon to take a solo in his first game, believing the freshman will panic and be embarrassed in front of everyone. Sean is shocked and subsequently humiliated when Devon takes the solo, upsetting Dr. Lee. As punishment, Dr. Lee orders Sean to clean the drums. This sets up some tension in the drumline which is exacerbated when Dr. Lee is told by President Wagner, the school's president, to change his focus from music to entertainment, otherwise he could potentially lose his funding. Lee does not want to give Devon more playing time because of his lack of respect and selfish attitude. Upon learning he lied on his college application about knowing how to read music, Devon is demoted to P4 by Dr. Lee until he learns, then later put back on P1 when Wagner pressures Dr. Lee to do so. However, after inflaming a melee with a visiting band at A&T's homecoming game after Devon plays on an opposing band member's drum (a serious insult in drumline mythos), Devon is expelled from the band by Dr. Lee. The fight also strains his relationship with Laila, as she is embarrassed to introduce him to her parents, who attended the game.


title: Drumline (film)

text: Devon contacts A&T's rival school Morris Brown College, to discuss playing for their band next season. Mr. Wade, Brown's band leader, says that Devon does not need to know how to read music and will likely get a full scholarship and a good position on the drumline. When Wade wants to know what Dr. Lee is planning for the BET Big Southern Classic (a large competition of college bands), Devon realizes Mr. Wade was merely trying to use him to steal A&T's performance plans, and that his heart and honor are still with the A&T band. Disgusted with both Mr. Wade and himself, he rejects the scholarship offer from the rival band and returns to A&T.


title: Drumline (film)

text: Though Devon is still not playing for the band, he refuses to give up his drumming. After receiving cassette tapes from his estranged father, Ray, Devon gets inspiration for new drum arrangements. As Devon goes to present these ideas to the band, he and Sean have a final confrontation that clears the air and they begin to work together. The two present their idea for an entrance cadence to Dr. Lee who decides to use them for the Classic. Devon helps the band prepare for the Classic and patches up his relationship with Laila. In appreciation for all his help getting the band ready for the Classic and impressed with Devon's maturing and growth, Dr. Lee tells Devon he can return to the band the following school year.


title: Drumline (film)

text: At the Classic, the bands are shown performing a mixture of popular songs. Morris Brown's band even gets rapper Petey Pablo to perform during their routine. A&T is not fazed by this and performs their mix of retro and current sounds. A tie results in the Morris Brown and A&T drumlines facing off against each other. Dr. Lee allows Devon to play for this face-off, showing his faith in Devon's improved character and in thanks for all the hard work he has done in getting the band ready for the Classic. Morris Brown goes first and A&T responds. Morris Brown's second cadence includes their snares moving forward and playing on the A&T drums (the same move that incited the fight at A&T's homecoming game), then throwing down their sticks. The A&T line manages to hold their composure in the face of the insult. They play their cadence and, in the middle, throw down their sticks, mimicking the Morris Brown actions, but then the entire line pulls out another set of sticks and continues playing. They end their routine in the faces of the Morris Brown drumline, but instead of playing on their drums, the line all drop their sticks onto the other drumline's drums. The judges award the win to A&T.


title: Drumline (film)

text: The film was released on VHS and in fullscreen and widescreen editions on DVD April 19, 2003. A "special edition" DVD version of the film was later released on December 6, 2011. The film was released in the Blu-ray format on January 27, 2009.


title: Drumline (film)

text: On Rotten Tomatoes, the film has an approval rating of 82% based on reviews from 84 critics. The site's consensus states: "Essentially a sports movie with drums, the energetic "Drumline" somehow manages to make the familiar seem fresh." It is tied for the #98 spot on the site's list of 100 Best Films of 2002. At "Metacritic", the film has averaged a 63 out of 100 rating from critics, based on 28 reviews. Audiences polled by CinemaScore gave the film a rare "A+" grade.


title: Drumline (film)

text: At "Yahoo! Movies", the film has been given a B average based on 14 reviews from critics, and a B− by over 30,000 users. The film was given 3½ stars at "Allmovie", where reviewer Josh Ralske gave positive note to the performances of the main cast and Stone's direction but still called the plot "formulaic."


title: Drumline (film)

text: Natasha Grant at "BlackFilm.com" called the movie "irresistible," "wonderfully crafted," and "fascinating." For the "Chicago Sun-Times", Roger Ebert gave the film 3 out of 4 stars, giving the film credit as being "entertaining" and "admirable." At the "Los Angeles Times", Kenneth Turan believed the film to be entertaining, although he states that the plot is "way-familiar." Writing for "The New York Times", A. O. Scott, while feeling the movie had a "skimpy, hectic plot," still believed the film to be "bouncy, boisterous and charming," and the play of the marching bands to be "downright thrilling." Mike Clark at "USA Today" awarded the film two and a half of four stars, feeling the film to be conventional but competent, and giving particular positive note to J. Anthony Brown, Orlando Jones, and Leonard Roberts performances. One of the films negative reviews came courtesy of David Levine at "FilmCritic.com". Giving the film 2.5 out of 5 stars, Levine called the screenplay "standard formula" and "predictable," and went on to say that it was unfunny as well as uninspiring. He said he was impressed by the precision and artistry of the marching bands.


title: Drumline (film)

text: The film's soundtrack was also executive produced by Dallas Austin. Released December 10, 2002, it reached the number 61 spot on Billboard's Top R&B/Hip-Hop Albums chart, number 10 on the Top Soundtracks list, and peaked at 133 on The Billboard 200 in 2003. The tracks "I Want a Girl Like You," "Blowin' Me Up (With Her Love)," and "Club Banger" were all released as singles, with JC Chasez' "Blowin' Me Up..." obtaining the highest level of success, reaching the number 24 spot on the Canadian Singles Chart, number 17 on the Top 40 Tracks chart, and number 14 on the Top 40 Mainstream chart.


title: Banach space

text: In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.


title: Banach space

text: Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.


title: Banach space

text: Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space."


title: Banach space

text: Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.


title: Banach space

text: formula_2 consisting of a vector space formula_3 over a scalar field formula_4 (where formula_4 is commonly formula_6 or formula_7) together with a distinguished


title: Banach space

text: A sequence formula_13 is called or or if for every real formula_14 there exists some index formula_15 such that


title: Banach space

text: The canonical metric formula_20 is called a if the pair formula_21 is a , which by definition means for every formula_13 in formula_23 there exists some formula_24 such that


title: Banach space

text: By definition, the normed space formula_2 is a if the norm induced metric formula_20 is a complete metric, or said differently, if formula_21 is a complete metric space.


title: Banach space

text: For any normed space formula_35 there exists an L-semi-inner product formula_36 on formula_3 such that formula_38 for all formula_24; in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.


title: Banach space

text: The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors.


title: Banach space

text: A normed space formula_3 is a Banach space if and only if each absolutely convergent series in formula_3 converges in formula_42


title: Banach space

text: The canonical metric formula_20 of a normed space formula_45 induces the usual metric topology formula_46 on formula_42 which is referred to as the canonical or norm induced topology.


title: Banach space

text: Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise.


title: Banach space

text: With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm formula_48 is always a continuous function with respect to the topology that it induces.


title: Banach space

text: The open and closed balls of radius formula_49 centered at a point formula_24 are, respectively, the sets


title: Banach space

text: Any such ball is a convex and bounded subset of formula_42 but a compact ball/neighborhood exists if and only if formula_3 is a finite-dimensional vector space.


title: Banach space

text: In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property.


title: Banach space

text: Using formula_57 shows that this norm-induced topology is translation invariant, which means that for any formula_24 and formula_59 the subset formula_60 is open (respectively, closed) in formula_3 if and only if this is true of its translation formula_62


title: Banach space

text: Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include:


title: Banach space

text: where formula_64 is a sequence in of positive real numbers that converges to formula_65 in formula_6 (such as formula_67 or formula_68 for instance).


title: Banach space

text: indexed by some subset formula_72 where every formula_73 is of the form formula_74 for some integer formula_75 (the closed ball can also be used instead of the open ball, although the indexing set formula_76 and radii formula_73 may need to be changed).


title: Banach space

text: Additionally, formula_76 can always be chosen to be countable if formula_3 is a , which by definition means that formula_3 contains some countable dense subset.


title: Banach space

text: The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space formula_81 of countably many copies of formula_6 (this homeomorphism need not be a linear map).


title: Banach space

text: Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including the separable Hilbert formula_83 sequence space formula_84 with its usual norm formula_85 where (in sharp contrast to finite−dimensional spaces) formula_84 is also homeomorphic to its unit formula_87


title: Banach space

text: There is a compact subset formula_60 of formula_84 whose convex hull formula_90 is closed and thus also compact (see this footnote for an example).


title: Banach space

text: However, like in all Banach spaces, the convex hull formula_91 of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general guaranteed that formula_91 will be compact whenever formula_60 is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of formula_94


title: Banach space

text: This norm-induced topology also makes formula_95 into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS formula_95 is a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with particular norm or metric (both of which are "forgotten"). This Hausdorff TVS formula_95 is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also , which by definition refers to any TVS whose its topology is induced by some (possibly unknown) norm.


title: Banach space

text: The open mapping theorem implies that if formula_98 are topologies on formula_3 that make both formula_100 and formula_101 into complete metrizable TVS (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if formula_102).


title: Banach space

text: So for example, if formula_103 are Banach spaces with topologies formula_104 and if one of these spaces has some open ball that is also an open subset of the other space (or equivalently, if one of formula_105 or formula_106 is continuous) then their topologies are identical and their norms are equivalent.


title: Banach space

text: Two norms, formula_107 and formula_108 on a vector space are said to be if they induce the same topology; this happens if and only if there exist positive real numbers formula_109 such that formula_110 for all formula_111 If formula_107 and formula_113 are two equivalent norms on a vector space formula_3 then formula_115 is a Banach space if and only if formula_116 is a Banach space.


title: Banach space

text: See this footnote for an example of a continuous norm on a Banach space that is equivalent to that Banach space's given norm.


title: Banach space

text: All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.


title: Banach space

text: A metric formula_117 on a vector space formula_3 is induced by a norm on formula_3 if and only if formula_117 is translation invariant and , which means that formula_121 for all scalars formula_122 and all formula_123 in which case the function formula_124 defines a norm on formula_3 and the canonical metric induced by formula_126 is equal to formula_127


title: Banach space

text: Suppose that formula_45 is a normed space and that formula_129 is the norm topology induced on formula_130 Suppose that formula_117 is metric on formula_3 such that the topology that formula_117 induces on formula_3 is equal to formula_135 If formula_117 is translation invariant then formula_45 is a Banach space if and only if formula_138 is a complete metric space.


title: Banach space

text: If formula_117 is translation invariant, then it may be possible for formula_45 to be a Banach space but for formula_138 to be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists complete metric formula_117 on formula_3 that induces the norm topology formula_129 on formula_42 then formula_45 is a Banach space.


title: Banach space

text: A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric.


title: Banach space

text: Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences formula_147 with the product topology).


title: Banach space

text: However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms.


title: Banach space

text: It is even possible for a Fréchet space to have a topology that is induced by a countable family of (such norms would necessarily be continuous)


title: Banach space

text: An example of such a space is the Fréchet space formula_148 whose definition can be found in the article on spaces of test functions and distributions.


title: Banach space

text: There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces.


title: Banach space

text: Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends on vector subtraction and the topology formula_129 that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology formula_129 (and even applies to TVSs that are even metrizable).


title: Banach space

text: Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space.


title: Banach space

text: If formula_100 is a metrizable topological vector space (such as any norm induced topology, for example), then formula_100 is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in formula_100 converges in formula_100 to some point of formula_3 (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).


title: Banach space

text: If formula_100 is a topological vector space whose topology is induced by (possibly unknown) norm (such spaces are called and they are characterized by being Hausdorff and having a bounded convex neighborhood of the origin), then formula_100 is a complete topological vector space if and only if formula_3 may be assigned a norm formula_126 that induces on formula_3 the topology formula_129 and also makes formula_45 into a Banach space.


title: Banach space

text: A Hausdorff locally convex topological vector space formula_3 is normable if and only if its strong dual space formula_164 is normable, in which case formula_164 is a Banach space (formula_164 denotes the strong dual space of formula_42 whose topology is a generalization of the dual norm-induced topology on the continuous dual space formula_168; see this footnote for more details).


title: Banach space

text: If formula_3 is a metrizable locally convex TVS, then formula_3 is normable if and only if formula_164 is a Fréchet–Urysohn space.


title: Banach space

text: This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.


title: Banach space

text: Every normed space can be isometrically embedded onto a dense vector subspace of Banach space, where this Banach space is called a of the normed space. This Hausdorff completion is unique up to isometric isomorphism.


title: Banach space

text: More precisely, for every normed space formula_42 there exist a Banach space formula_173 and a mapping formula_174 such that formula_175 is an isometric mapping and formula_176 is dense in formula_177 If formula_178 is another Banach space such that there is an isometric isomorphism from formula_3 onto a dense subset of formula_180 then formula_178 is isometrically isomorphic to formula_177


title: Banach space

text: This Banach space formula_173 is the Hausdorff of the normed space formula_130 The underlying metric space for formula_173 is the same as the metric completion of formula_42 with the vector space operations extended from formula_3 to formula_177 The completion of formula_3 is sometimes denoted by formula_190


title: Banach space

text: If formula_3 and formula_173 are normed spaces over the same ground field formula_193 the set of all continuous formula_4-linear maps formula_174 is denoted by formula_196 In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space formula_3 to another normed space is continuous if and only if it is bounded on the closed unit ball of formula_130 Thus, the vector space formula_199 can be given the operator norm


title: Banach space

text: For formula_173 a Banach space, the space formula_199 is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space formula_203 reappears as a natural bifunctor.


title: Banach space

text: If formula_3 is a Banach space, the space formula_205 forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.


title: Banach space

text: If formula_3 and formula_173 are normed spaces, they are isomorphic normed spaces if there exists a linear bijection formula_174 such that formula_175 and its inverse formula_210 are continuous. If one of the two spaces formula_3 or formula_173 is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces formula_3 and formula_173 are isometrically isomorphic if in addition, formula_175 is an isometry, that is, formula_216 for every formula_27 in formula_130 The Banach–Mazur distance formula_219 between two isomorphic but not isometric spaces formula_3 and formula_173 gives a measure of how much the two spaces formula_3 and formula_173 differ.


title: Banach space

text: Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is formula_6 or formula_7) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.


title: Banach space

text: If formula_226 is a subadditive function (such as a norm, a sublinear function, or real linear functional), then formula_227 is continuous at the origin if and only if formula_227 is uniformly continuous on all of formula_3; and if in addition formula_230 then formula_227 is continuous if and only if its absolute value formula_232 is continuous, which happens if and only if formula_233 is an open subset of formula_130


title: Banach space

text: And very importantly for applying the Hahn–Banach theorem, a linear functional formula_227 is continuous if and only if this is true of its real part formula_236 and moreover, formula_237 and the real part formula_236 completely determines formula_239 which is why the Hahn–Banach theorem is often stated only for real linear functionals.


title: Banach space

text: Also, a linear functional formula_227 on formula_3 is continuous if and only if the seminorm formula_242 is continuous, which happens if and only if there exists a continuous seminorm formula_243 such that formula_244; this last statement involving the linear functional formula_227 and seminorm formula_107 is encountered in many versions of the Hahn–Banach theorem.


title: Banach space

text: The Cartesian product formula_247 of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as


title: Banach space

text: which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product formula_247 (or the direct sum formula_250) is complete if and only if the two factors are complete.


title: Banach space

text: If formula_251 is a closed linear subspace of a normed space formula_42 there is a natural norm on the quotient space formula_253


title: Banach space

text: The quotient formula_255 is a Banach space when formula_3 is complete. The quotient map from formula_3 onto formula_253 sending formula_24 to its class formula_260 is linear, onto and has norm formula_261 except when formula_262 in which case the quotient is the null space.


title: Banach space

text: The closed linear subspace formula_251 of formula_3 is said to be a complemented subspace of formula_3 if formula_251 is the range of a surjective bounded linear projection formula_267 In this case, the space formula_3 is isomorphic to the direct sum of formula_251 and formula_270 the kernel of the projection formula_271


title: Banach space

text: Suppose that formula_3 and formula_173 are Banach spaces and that formula_274 There exists a canonical factorization of formula_175 as


title: Banach space

text: where the first map formula_277 is the quotient map, and the second map formula_278 sends every class formula_279 in the quotient to the image formula_280 in formula_281 This is well defined because all elements in the same class have the same image. The mapping formula_278 is a linear bijection from formula_283 onto the range formula_284 whose inverse need not be bounded.


title: Banach space

text: Basic examples of Banach spaces include: the Lp spaces formula_285 and their special cases, the sequence spaces formula_286 that consist of scalar sequences indexed by natural numbers formula_287; among them, the space formula_288 of absolutely summable sequences and the space formula_289 of square summable sequences; the space formula_290 of sequences tending to zero and the space formula_291 of bounded sequences; the space formula_292 of continuous scalar functions on a compact Hausdorff space formula_293 equipped with the max norm,


title: Banach space

text: According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some formula_295 For every separable Banach space formula_42 there is a closed subspace formula_251 of formula_288 such that formula_299


title: Banach space

text: Any Hilbert space serves as an example of a Banach space. A Hilbert space formula_300 on formula_301 is complete for a norm of the form


title: Banach space

text: The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to formula_285 spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.


title: Banach space

text: A Banach algebra is a Banach space formula_307 over formula_308 or formula_309 together with a structure of algebra over formula_4, such that the product map formula_311 is continuous. An equivalent norm on formula_307 can be found so that formula_313 for all formula_314


title: Banach space

text: If formula_3 is a normed space and formula_4 the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from formula_3 into formula_193 or continuous linear functionals.


title: Banach space

text: Since formula_4 is a Banach space (using the absolute value as norm), the dual formula_168 is a Banach space, for every normed space formula_130


title: Banach space

text: In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.


title: Banach space

text: An important special case is the following: for every vector formula_27 in a normed space formula_42 there exists a continuous linear functional formula_227 on formula_3 such that


title: Banach space

text: When formula_27 is not equal to the formula_353 vector, the functional formula_227 must have norm one, and is called a norming functional for formula_355


title: Banach space

text: The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane.


title: Banach space

text: The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.


title: Banach space

text: A subset formula_60 in a Banach space formula_3 is total if the linear span of formula_60 is dense in formula_130 The subset formula_60 is total in formula_3 if and only if the only continuous linear functional that vanishes on formula_60 is the formula_353 functional: this equivalence follows from the Hahn–Banach theorem.


title: Banach space

text: If formula_3 is the direct sum of two closed linear subspaces formula_251 and formula_366 then the dual formula_168 of formula_3 is isomorphic to the direct sum of the duals of formula_251 and formula_19


title: Banach space

text: If formula_251 is a closed linear subspace in formula_42 one can associate the formula_251 in the dual,


title: Banach space

text: The orthogonal formula_375 is a closed linear subspace of the dual. The dual of formula_251 is isometrically isomorphic to formula_377


title: Banach space

text: When formula_380 is separable, the above criterion for totality can be used for proving the existence of a countable total subset in formula_130


title: Banach space

text: The weak topology on a Banach space formula_3 is the coarsest topology on formula_3 for which all elements formula_384 in the continuous dual space formula_168 are continuous.


title: Banach space

text: It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.


title: Banach space

text: A norm-continuous linear map between two Banach spaces formula_3 and formula_173 is also weakly continuous, that is, continuous from the weak topology of formula_3 to that of formula_177


title: Banach space

text: If formula_3 is infinite-dimensional, there exist linear maps which are not continuous. The space formula_391 of all linear maps from formula_3 to the underlying field formula_4 (this space formula_391 is called the algebraic dual space, to distinguish it from formula_168 also induces a topology on formula_3 which is finer than the weak topology, and much less used in functional analysis.


title: Banach space

text: On a dual space formula_397 there is a topology weaker than the weak topology of formula_397 called weak* topology.


title: Banach space

text: It is the coarsest topology on formula_168 for which all evaluation maps formula_400 where formula_27 ranges over formula_42 are continuous.


title: Banach space

text: The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces.


title: Banach space

text: When formula_3 is separable, the unit ball formula_404 of the dual is a metrizable compact in the weak* topology.


title: Banach space

text: The dual of formula_290 is isometrically isomorphic to formula_288: for every bounded linear functional formula_227 on formula_408 there is a unique element formula_409 such that


title: Banach space

text: The dual of Lebesgue space formula_413 is isometrically isomorphic to formula_414 when formula_415 and formula_416


title: Banach space

text: defines a continuous linear functional formula_420 on formula_421The Riesz representation theorem states that every continuous linear functional on formula_300 is of the form formula_420 for a uniquely defined vector formula_417 in formula_421


title: Banach space

text: The mapping formula_426 is an antilinear isometric bijection from formula_300 onto its dual formula_428


title: Banach space

text: When formula_336 is a compact Hausdorff topological space, the dual formula_430 of formula_292 is the space of Radon measures in the sense of Bourbaki.


title: Banach space

text: The subset formula_432 of formula_430 consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of formula_434


title: Banach space

text: The set of Dirac measures on formula_293 equipped with the w*-topology, is homeomorphic to formula_436


title: Banach space

text: The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between formula_292 and formula_440 is formula_441


title: Banach space

text: In the commutative Banach algebra formula_443 the maximal ideals are precisely kernels of Dirac measures on formula_293


title: Banach space

text: More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology.


title: Banach space

text: In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual formula_446


title: Banach space

text: Not every unital commutative Banach algebra is of the form formula_292 for some compact Hausdorff space formula_436 However, this statement holds if one places formula_292 in the smaller category of commutative C*-algebras.


title: Banach space

text: Gelfand's representation theorem for commutative C*-algebras states that every commutative unital "C"*-algebra formula_307 is isometrically isomorphic to a formula_292 space.


title: Banach space

text: The Hausdorff compact space formula_336 here is again the maximal ideal space, also called the spectrum of formula_307 in the C*-algebra context.


title: Banach space

text: If formula_3 is a normed space, the (continuous) dual formula_455 of the dual formula_380 is called ', or ' of formula_130


title: Banach space

text: This defines formula_460 as a continuous linear functional on formula_397 that is, an element of formula_462 The map formula_463 is a linear map from formula_3 to formula_462


title: Banach space

text: As a consequence of the existence of a norming functional formula_227 for every formula_467 this map formula_468 is isometric, thus injective.


title: Banach space

text: For example, the dual of formula_469 is identified with formula_470 and the dual of formula_288 is identified with formula_472 the space of bounded scalar sequences.


title: Banach space

text: Under these identifications, formula_468 is the inclusion map from formula_290 to formula_475 It is indeed isometric, but not onto.


title: Banach space

text: Being the dual of a normed space, the bidual formula_455 is complete, therefore, every reflexive normed space is a Banach space.


title: Banach space

text: Using the isometric embedding formula_479 it is customary to consider a normed space formula_3 as a subset of its bidual.


title: Banach space

text: When formula_3 is a Banach space, it is viewed as a closed linear subspace of formula_462 If formula_3 is not reflexive, the unit ball of formula_3 is a proper subset of the unit ball of formula_462


title: Banach space

text: The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual.


title: Banach space

text: In other words, for every formula_486 in the bidual, there exists a net formula_487 in formula_3 so that


title: Banach space

text: On the other hand, elements of the bidual of formula_288 that are not in formula_288 cannot be weak*-limit of in formula_470 since formula_288 is weakly sequentially complete.


title: Banach space

text: Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the Baire category theorem.


title: Banach space

text: According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors.


title: Banach space

text: Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.


title: Banach space

text: It can be extended for example to the case where formula_3 is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood formula_69 of formula_353 in formula_3 such that all formula_175 in formula_500 are uniformly bounded on formula_501


title: Banach space

text: This result is a direct consequence of the preceding "Banach isomorphism theorem" and of the canonical factorization of bounded linear maps.


title: Banach space

text: This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from formula_503 onto formula_3 sending formula_505 to the sum formula_506


title: Banach space

text: Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space formula_3 onto the Banach space formula_510 then formula_173 is reflexive.


title: Banach space

text: Indeed, if the dual formula_512 of a Banach space formula_173 is separable, then formula_173 is separable.


title: Banach space

text: If formula_3 is reflexive and separable, then the dual of formula_168 is separable, so formula_168 is separable.


title: Banach space

text: Hilbert spaces are reflexive. The formula_285 spaces are reflexive when formula_519 More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem.


title: Banach space

text: In these examples of non-reflexive spaces formula_42 the bidual formula_455 is "much larger" than formula_130


title: Banach space

text: Namely, under the natural isometric embedding of formula_3 into formula_455 given by the Hahn–Banach theorem, the quotient formula_526 is infinite-dimensional, and even nonseparable.


title: Banach space

text: However, Robert C. James has constructed an example of a non-reflexive space, usually called ""the James space"" and denoted by formula_527 such that the quotient formula_528 is one-dimensional.


title: Banach space

text: When formula_3 is reflexive, it follows that all closed and bounded convex subsets of formula_3 are weakly compact.


title: Banach space

text: In a Hilbert space formula_418 the weak compactness of the unit ball is very often used in the following way: every bounded sequence in formula_300 has weakly convergent subsequences.


title: Banach space

text: Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.


title: Banach space

text: For example, every convex continuous function on the unit ball formula_534 of a reflexive space attains its minimum at some point in formula_535


title: Banach space

text: As a special case of the preceding result, when formula_3 is a reflexive space over formula_537 every continuous linear functional formula_227 in formula_168 attains its maximum formula_540 on the unit ball of formula_130


title: Banach space

text: On every non-reflexive Banach space formula_42 there exist continuous linear functionals that are not "norm-attaining".


title: Banach space

text: However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual formula_168 of formula_130


title: Banach space

text: A sequence formula_545 in a Banach space formula_3 is weakly convergent to a vector formula_24 if formula_548 converges to formula_549 for every continuous linear functional formula_227 in the dual formula_551 The sequence formula_545 is a weakly Cauchy sequence if formula_548 converges to a scalar limit formula_554 for every formula_227 in formula_551


title: Banach space

text: A sequence formula_557 in the dual formula_168 is weakly* convergent to a functional formula_559 if formula_560 converges to formula_549 for every formula_27 in formula_130


title: Banach space

text: Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.


title: Banach space

text: When the sequence formula_545 in formula_3 is a weakly Cauchy sequence, the limit formula_566 above defines a bounded linear functional on the dual formula_397 that is, an element formula_566 of the bidual of formula_42 and formula_566 is the limit of formula_545 in the weak*-topology of the bidual.


title: Banach space

text: The Banach space formula_3 is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in formula_130


title: Banach space

text: An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the formula_353 vector.


title: Banach space

text: The unit vector basis of formula_286 for formula_576 or of formula_408 is another example of a weakly null sequence, that is, a sequence that converges weakly to formula_578


title: Banach space

text: For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to formula_578


title: Banach space

text: Weakly Cauchy sequences in formula_288 are weakly convergent, since formula_582-spaces are weakly sequentially complete.


title: Banach space

text: Actually, weakly convergent sequences in formula_288 are norm convergent. This means that formula_288 satisfies Schur's property.


title: Banach space

text: Weakly Cauchy sequences and the formula_288 basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.


title: Banach space

text: By the Goldstine theorem, every element of the unit ball formula_587 of formula_588 is weak*-limit of a net in the unit ball of formula_130 When formula_3 does not contain formula_470 every element of formula_587 is weak*-limit of a in the unit ball of formula_130


title: Banach space

text: When the Banach space formula_3 is separable, the unit ball of the dual formula_397 equipped with the weak*-topology, is a metrizable compact space formula_293 and every element formula_597 in the bidual formula_588 defines a bounded function on formula_336:


title: Banach space

text: This function is continuous for the compact topology of formula_336 if and only if formula_597 is actually in formula_42 considered as subset of formula_462


title: Banach space

text: By the preceding result of Odell and Rosenthal, the function formula_597 is the pointwise limit on formula_336 of a sequence formula_609 of continuous functions on formula_293 it is therefore a first Baire class function on formula_436


title: Banach space

text: When formula_3 is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences.


title: Banach space

text: The weak topology of a Banach space formula_3 is metrizable if and only if formula_3 is finite-dimensional. If the dual formula_380 is separable, the weak topology of the unit ball of formula_3 is metrizable.


title: Banach space

text: Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.


title: Banach space

text: A Banach space formula_3 is reflexive if and only if each bounded sequence in formula_3 has a weakly convergent subsequence.


title: Banach space

text: A weakly compact subset formula_307 in formula_288 is norm-compact. Indeed, every sequence in formula_307 has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of formula_606


title: Banach space

text: A Schauder basis in a Banach space formula_3 is a sequence formula_625 of vectors in formula_3 with the property that for every vector formula_467 there exist defined scalars formula_628 depending on formula_629 such that


title: Banach space

text: Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.


title: Banach space

text: It follows from the Banach–Steinhaus theorem that the linear mappings formula_631 are uniformly bounded by some constant formula_632


title: Banach space

text: Let formula_633 denote the coordinate functionals which assign to every formula_27 in formula_3 the coordinate formula_636 of formula_27 in the above expansion.


title: Banach space

text: They are called biorthogonal functionals. When the basis vectors have norm formula_261 the coordinate functionals formula_639 have norm formula_640 in the dual of formula_130


title: Banach space

text: The question of whether the disk algebra formula_316 has a basis remained open for more than forty years, until Bočkarev showed in 1974 that formula_316 admits a basis constructed from the Franklin system.


title: Banach space

text: Since every vector formula_27 in a Banach space formula_3 with a basis is the limit of formula_651 with formula_652 of finite rank and uniformly bounded, the space formula_3 satisfies the bounded approximation property.


title: Banach space

text: The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.


title: Banach space

text: Robert C. James characterized reflexivity in Banach spaces with a basis: the space formula_3 with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.


title: Banach space

text: Let formula_3 and formula_173 be two formula_4-vector spaces. The tensor product formula_659 of formula_3 and formula_173 is a formula_4-vector space formula_178 with a bilinear mapping formula_664 which has the following universal property:


title: Banach space

text: The image under formula_175 of a couple formula_671 in formula_247 is denoted by formula_673 and called a simple tensor.


title: Banach space

text: There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.


title: Banach space

text: In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces formula_3 and formula_173 is the formula_678 of the algebraic tensor product formula_659 equipped with the projective tensor norm, and similarly for the injective tensor product formula_680


title: Banach space

text: where formula_336 is a compact Hausdorff space, formula_683 the Banach space of continuous functions from formula_336 to formula_173 and formula_686 the space of Bochner-measurable and integrable functions from formula_687 to formula_510 and where the isomorphisms are isometric.


title: Banach space

text: The two isomorphisms above are the respective extensions of the map sending the tensor formula_689 to the vector-valued function formula_690


title: Banach space

text: Let formula_3 be a Banach space. The tensor product formula_692 is identified isometrically with the closure in formula_327 of the set of finite rank operators.


title: Banach space

text: When formula_3 has the approximation property, this closure coincides with the space of compact operators on formula_130


title: Banach space

text: obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when formula_173 is the dual of formula_130


title: Banach space

text: Grothendieck conjectured that formula_678 and formula_705 must be different whenever formula_3 and formula_173 are infinite-dimensional Banach spaces.


title: Banach space

text: Pisier constructed an infinite-dimensional Banach space formula_3 such that formula_709 and formula_710 are equal. Furthermore, just as Enflo's example, this space formula_3 is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space formula_712 does not have the approximation property.


title: Banach space

text: A necessary and sufficient condition for the norm of a Banach space formula_3 to be associated to an inner product is the parallelogram identity:


title: Banach space

text: It follows, for example, that the Lebesgue space formula_413 is a Hilbert space only when formula_715


title: Banach space

text: If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:


title: Banach space

text: For complex scalars, defining the inner product so as to be formula_7-linear in formula_629 antilinear in formula_719 the polarization identity gives:


title: Banach space

text: To see that the parallelogram law is sufficient, one observes in the real case that formula_721 is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and formula_722 The parallelogram law implies that formula_721 is additive in formula_355


title: Banach space

text: Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available.


title: Banach space

text: The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant formula_725: Kwapień proved that if


title: Banach space

text: for every integer formula_18 and all families of vectorsformula_728 then the Banach space formula_3 is isomorphic to a Hilbert space.


title: Banach space

text: In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces.


title: Banach space

text: Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer formula_733 any finite-dimensional normed space, with dimension sufficiently large compared to formula_733 contains subspaces nearly isometric to the formula_18-dimensional Euclidean space.


title: Banach space

text: The next result gives the solution of the so-called . An infinite-dimensional Banach space formula_3 is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to formula_289 is homogeneous, and Banach asked for the converse.


title: Banach space

text: An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces.


title: Banach space

text: The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space formula_3 contains, either a subspace formula_173 with unconditional basis, or a hereditarily indecomposable subspace formula_180 and in particular, formula_178 is not isomorphic to its closed hyperplanes.


title: Banach space

text: If formula_3 is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis, that formula_3 is isomorphic to formula_744


title: Banach space

text: If formula_174 is an isometry from the Banach space formula_3 onto the Banach space formula_173 (where both formula_3 and formula_173 are vector spaces over formula_6), then the Mazur–Ulam theorem states that formula_175 must be an affine transformation.


title: Banach space

text: In particular, if formula_752 this is formula_175 maps the zero of formula_3 to the zero of formula_510 then formula_175 must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.


title: Banach space

text: Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.


title: Banach space

text: Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.


title: Banach space

text: When two compact Hausdorff spaces formula_757 and formula_758 are homeomorphic, the Banach spaces formula_759 and formula_760 are isometric. Conversely, when formula_757 is not homeomorphic to formula_762 the (multiplicative) Banach–Mazur distance between formula_759 and formula_760 must be greater than or equal to formula_765 see above the results by Amir and Cambern.


title: Banach space

text: Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:


title: Banach space

text: Every countably infinite compact formula_336 is homeomorphic to some closed interval of ordinal numbers


title: Banach space

text: The Banach space formula_292 is then isometric to . When formula_770 are two countably infinite ordinals, and assuming formula_771 the spaces and are isomorphic if and only if .


title: Banach space

text: Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details.


title: Banach space

text: The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.


title: Banach space

text: The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.


title: Banach space

text: Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions formula_773 or the space of all distributions on formula_537 are complete but are not normed vector spaces and hence not Banach spaces.


title: Banach space

text: In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.


title: House dust mite

text: House dust mites (HDM, or simply dust mites) are various species of acariform mites belonging to the family Pyroglyphidae that are found in association with dust in dwellings. They are known for causing allergies.


title: House dust mite

text: House dust mites, due to their very small size and translucent bodies, are barely visible to the unaided eye. A typical house dust mite measures 0.2–0.3 mm in length. The body of the house dust mite has a striated cuticle.


title: House dust mite

text: They feed on skin flakes from humans and other animals, and on some mold. "Dermatophagoides farinae" fungal food choices in 16 tested species commonly found in homes was observed "in vitro" to be "Alternaria alternata", "Cladosporium sphaerospermum", and "Wallemia sebi", and they disliked "Penicillium chrysogenum", "Aspergillus versicolor", and "Stachybotrys chartarum".


title: House dust mite

text: The predators of dust mites are other allergenic mites ("Cheyletiella"), silverfish and pseudoscorpions.


title: House dust mite

text: The average life cycle for a house dust mite is 65–100 days. A mated female house dust mite can live up to 70 days, laying 60 to 100 eggs in the last five weeks of her life. In a 10-week life span, a house dust mite will produce approximately 2,000 fecal particles and an even larger number of partially digested enzyme-covered dust particles.


title: House dust mite

text: Dust mites are found worldwide, but are more common in humid regions. The species "Blomia tropicalis" is typically found only in tropical or subtropical regions. Detectable dust mite allergen was found in the beds of about 84% of surveyed United States homes. In Europe, detectable Der p 1 or Der f 1 allergen was found in 68% of surveyed homes.


title: Mankatha

text: Mankatha () is a 2011 Indian Tamil-language action crime film written and directed by Venkat Prabhu. It stars Ajith Kumar in his 50th film, along with an ensemble cast including Arjun Sarja, Trisha Krishnan, Lakshmi Rai, Anjali, Andrea Jeremiah, Vaibhav Reddy, Premji Amaren, and Mahat Raghavendra . It was produced by Dhayanidhi Alagiri's Cloud Nine Movies while Yuvan Shankar Raja composed the musical score and soundtrack, with Sakthi Saravanan working as the cinematographer and the duo Praveen K. L. and N. B. Srikanth as editors. The story, set in Mumbai, revolves around a heist of cricket betting money, executed by a gang of four thieves, who are joined by a fifth unknown man, and its aftermath.


title: Mankatha

text: The film was formally launched in August 2010, with its principal photography beginning on 25 October 2010. Filming was held for more than eight months and took place primarily across Chennai, the Dharavi slum in Mumbai and Bangkok, Thailand. Following speculations regarding the film's release, Sun Pictures acquired the theatrical rights and distributed the film via Raadhika's Radaan Mediaworks. "Mankatha" released on 31 August 2011 worldwide to generally positive reviews and grossed the second biggest opening of all time after "Enthiran" at the time of release.


title: Mankatha

text: Vinayak Mahadev, an ACP of Maharashtra Police, is suspended for having saved smuggler Faizal from encounter killing and helping him escape. Meanwhile, a police officer commits suicide because of the leakage of his IPL gambling plans, who is revealed to be Kamal Ekambaram. ACP Prithviraj takes charge to end the betting scandals in IPL cricket in Mumbai. Prithvi then reveals that Kamal faked his death in a secret mission to draw attention to the betting scandals and returns under the name Praveen Kumar. Arumuga Chettiyar, an influential illegal business dealer and Faizal's boss, owns "Golden Theatres" in Mumbai, which has been converted into a gambling den and forms the front for all his illegal businesses. Chettiyar uses his links with dons in Mumbai and tries to route through his old theatre, a cash of over to be used in betting.


title: Mankatha

text: Vinayak is introduced to Chettiyar through his girlfriend Sanjana. Sanjana is in love with Vinayak, but he pretends to love her. Sumanth, a goon working for Chettiyar, hatches a conspiracy to rob the money in the company of his friends: Ganesh, a local Sub-Inspector; Mahat, who owns a bar in Mumbai; and Mahat's friend Prem, an IIT graduate. Vinayak befriends the boys at Sumanth's marriage with Suchithra. One late evening, Vinayak meets Prem, who becomes inebriated by him and reveals their heist plan. Vinayak starts spying on them and confronts them on the day of the planned heist, stepping in. The four take him in, promising him a fifth of the share. Vinayak has other plans. He wants to kill his accomplices and take the entire amount. He promises to help them and divide it between them. After looting the money, they leave the money in an abandoned godown.


title: Mankatha

text: Later, all of them celebrate the turn of events at Mahat's bar, but Sumanth is identified at the party by Faizal and is later caught by him. Sumanth is cornered by Chettiyar, who orders Faizal to kill him for his treachery, but is rescued in time by Ganesh and Vinayak, and the trio escape from the hideout, taking Chettiyar hostage. While driving back to the godown, Vinayak finds Sanjana on the way and brutally shoves Chettiyar out of the vehicle in front of her. Sanjana engulfs in grief when she learns of Vinayak's true intentions. Upon reaching the godown, they discover that Mahat and Prem had escaped with the cash and are accompanied by Sona. The three are then confronted by Faizal and Chettiyar's men, but manage to evade them and get on the run. Sumanth later turns against Vinayak when he discovers that his wife had been kidnapped by Chettiyar.


title: Mankatha

text: After a brief scuffle, Sumanth runs into Prithvi, who takes him to custody and rescues his wife on the condition that he turns approver and divulge everything. He is killed when Prithvi's wife Sabitha is kidnapped and threatened by Vinayak. Vinayak learns of Mahat and Prem's whereabouts through Ganesh, and along with him, starts pursuing them. Prithvi and the others follow suit, and all of them are holed up in a highway resort with the money. A sequence of events lead to the murders of the gang members one by one — Mahat is killed by Sona, who gets killed by Vinayak and Prem gets killed by Prithvi — with Ganesh and Vinayak remaining alive. A final fight ensues between Vinayak and Prithvi. At the final moment of the fight, Praveen throws a gun at Prithvi, who shoots Vinayak, and a huge explosion rocks the shack, seemingly ending the fight.


title: Mankatha

text: After several days, the police gets information about Ganesh to be living in Thailand. Praveen arrives there but comes across Vinayak. Praveen confronts him and calls up Prithvi to inform him of Vinayak's presence, but it is revealed that Prithvi and Vinayak are best friends since their college days and took police training together. They had come to know about the betting money scheme by Chettiyar and operated the plan together (including Vinayak's faked death). Ganesh had also been killed by Vinayak as a part of the plan, and Prithvi and Vinayak escaped the explosion with the , each taking as their share. Prithvi informs Vinayak that their money is safe in the Bank of England and asks him to deal with Praveen. Vinayak then snatches Praveen's gun and holds him at gunpoint, saying "Game over".


title: Mankatha

text: Following the release of his film "Aasal" in February 2010, Ajith Kumar was signed by Dhayanidhi Alagiri's Cloud Nine Movies for a project touted to be directed by Gautham Vasudev Menon. However, as Ajith participated in the 2010 season of the FIA Formula Two Championship, Gautham Vasudev Menon, unwilling to wait, decided to shelve the film and opted to focus on another project, which prompted Venkat Prabhu, who was keen on making a multi-starrer film featuring top stars, to sign the actor in his next film, after the director saw moderate success with his previous film, "Goa". Prabhu had written three scripts, out of which the actor chose "Mankatha", in which he would portray a character "with grey shades". Venkat Prabhu later disclosed that Ajith was never considered for the role when he wrote the first draft of the script but that he kept his "usual gang of boys in mind" while creating the characters. Vaibhav Reddy suggested that a "big hero" should play the protagonist, with actor Vivek Oberoi and Sathyaraj being considered first for the role. Ajith had called Prabhu at that time and expressed interest in performing a role similar to The Joker character played by Heath Ledger in the 2008 American superhero film "The Dark Knight". With the protagonist role in "Mankatha" incidentally being such a character, Ajith immediately accepted the role, turning the film into a high-profile production. Prabhu further emphasised that the script had been altered due to Ajith's entry and he had incorporated "certain elements" that Ajith's fans would expect in a film and also said that "I approached Mankatha as a fan and asked myself how I would want to see Ajith sir on screen and then set out".


title: Mankatha

text: The film officially commenced on 2 August 2010 with the formal launch and a simple pooja held at the AVM Studios, Chennai, coinciding with Ajith's 18th anniversary of his entry into the film industry. The film's title, initially being "Mangaatha", derived from a popular Indian traditional card game, underwent a minor change in its spelling, due to numerological reasons. During the pre-production stage, while Prabhu was still working on the scriptment, sources claimed the film to be on the lines of Steven Soderbergh's heist film "Ocean Eleven" (2001). The story was later reported to revolve around a Mafia gang gambling during the Indian Premier League (IPL) cricket season. In June 2011, reports emerged that the film was a remake of the 2008 Hindi film "Jannat" that was based on match fixing. However, Dhayanidhi and Venkat Prabhu quickly denied the news and assured that "Mankatha" was original. Upon completion of filming, Prabhu named it "his favourite film so far" and "close to my heart".


title: Mankatha

text: The film was supposed to be a "multi-starrer", with several leading Indian actors expected to appear in it, which is a rare occurrence and would be the first of its kind in the Tamil film industry. Telugu actor Nagarjuna was first approached to essay a powerful character as a CBI officer, being initially confirmed by the director during the launch of the film. He had also conveyed interest in remaking the film into Telugu and playing Ajith's role in return, but since he could not adjust his call sheet, he was forced to pull out, with Arjun replacing him in November 2010. In an interview from August 2010, Venkat Prabhu had affirmed that Mohan Babu's son, Manoj Manchu, Ganesh Venkatraman, Venkat Prabhu's younger brother, Premji Amaren, who had been part of all his brothers' films, and a newcomer Mahat Raghavendra, a childhood friend of producer Dhayanidhi Alagiri, were signed to portray Ajith's sidekicks in the film. However, Manoj Manchu couldn't accept the offer due to a shoulder injury and was eventually replaced by Vaibhav Reddy, appearing in the third consecutive Venkat Prabhu film, while in December 2010, sources revealed that another newcomer Ashwin Kakumanu, who previously was seen in "Nadunisi Naaygal", was roped in for the fourth role, replacing Ganesh Venkatraman. Nandha Durairaj later stated that he was offered the role as well, but had to reject it since he was busy shooting for "Vellore Maavattam". Prasanna was also considered for the film only to reject due to busy schedules. Jai in 2013 revealed that he was initially roped in to play the police officer role but that Venkat Prabhu replaced him with Arjun after Ajith became part of the project.


title: Mankatha

text: Prabhu termed the film as "male-oriented", attaching less importance and significance to the female characters in the film. Early reports suggested that Samantha Ruth Prabhu, Kajal Aggarwal and Anushka Shetty were initially approached for the lead female roles, while in July 2010, reports surfaced that Shriya Saran and Neetu Chandra, and Lakshmi Rai were supposedly signed for the roles. However, the following month Neetu Chandra opted out of the film, refusing the project due to unavailability of dates. Subsequently, Trisha was signed to portray Sanjana, Ajith's love interest, pairing with him for the third time, with Lakshmi Rai being confirmed later, who was signed on to play an important and "lengthy role". Venkat Prabhu revealed that changes in the script resulted in changes of the female characters' personalities, clarifiying that Trisha was not the replacement for Neetu Chandra and that the characters offered to both were different, while adding that Trisha's role was specifically written for her. Prabhu further stated that Lakshmi Rai was first chosen to play Ashwin's pair, which was dropped when the screenplay was altered, and that she was eventually offered the role of Sona, resolving that Rai was not given a choice to select between the roles of Sanjana and Sona, as the actress had claimed post the film's release. Earlier, reports had claimed that model and actress Jacqueline Fernandez was also roped in for a guest role. In November 2010, Sneha was reported to be added to the cast to be paired opposite Arjun, however the role was later finalised with Andrea Jeremiah portraying that character. She was also expected to perform a song for the soundtrack album.


title: Mankatha

text: Other additions to the cast in the following months included Subbu Panchu, who rose to fame with his appearance in "Boss Engira Bhaskaran" and would essay a police officer character, Jayaprakash, enacting also a character with negative shades as the father of Trisha's character, and Anjali, playing Vaibhav's pair. Sources claimed that Venkat Prabhu himself would also enact a pivotal role in the film, while reports suggesting that Vijay would appear in a cameo role were dismissed by the producer, who clarified that Vijay Vasanth would appear in a pivotal role. Concerning the film crew, Venkat Prabhu renewed his previous associations with his cousin Yuvan Shankar Raja, for the background score and soundtrack of "Mankatha", Sakthi Saravanan, who would handle the cinematography, and Praveen K. L., who along with N. B. Srikanth, would take care of the editing. Vasuki Bhaskar and Kalyan remained the costume designer and the main choreographer, respectively, with Shoby joining the latter for a couple of songs, while Selva was assigned as the stunt coordinator.


title: Mankatha

text: The film was launched on 2 August 2010 at AVM Studios in Vadapalani, Chennai in a simple manner, following which the film's shooting commenced with the principal photography. The first schedule of filming was supposed to begin by early September, but due to pre-production works and since the principal cast was not decided yet, the shooting got delayed further, finally commencing on 25 October in Chennai. Several days earlier, a test shoot was conducted with Premji Amaren, Mahat Raghavendra and Vaibhav Reddy taking part in it. Following shoots, involving Ajith, Trisha and Premji, along the Rajiv Gandhi Salai (OMR), and at Ajith's home, a duet song, picturized on Ajith and Trisha, featuring CGI special effects, was filmed in early November, in a Chennai studio nearby the East Coast Road. From 10 November onwards, the "introduction" song was shot for five days in Bangkok, Thailand, with Ajith, Lakshmi Rai and some foreigners participating.


title: Mankatha

text: The film's second schedule was planned to begin on 6 December 2010 in a studio in Chennai, which was slightly delayed due to heavy rain, and started couple of days later. This led to speculation that the film had been shelved due to financial constraints, which was quickly denied by Venkat Prabhu. During the schedule, all important stunt sequences were canned at Binny Mills in Perambur, while simultaneously a grand set, resembling the Dharavi slum in Mumbai was erected in a Chennai studio. Ajith also performed one of the action choreographies with the use of a body mounted camera, weighing around 30 kg. In late December, the third song, a "high-spirited peppy number", was shot for five days, with Shobi choreographing the steps. An item number, titled "Machi, Open The Bottle", it featured actresses Debi Dutta and Kainaat Arora dancing to the song along with Ajith and the rest of the gang The schedule was wrapped up by early February, with which approximately fifty per cent of the film was reportedly completed.


title: Mankatha

text: The remaining part of the film was supposedly to be shot during the third and last schedule to be held in Mumbai, which was to start in late February. However, sourced clarified that the subsequent schedule, too, would be held in Chennai only, with filming being carried on at the Padmanabha Theatre in North Chennai. In late March, the crew eventually moved to Mumbai, where the filming was held for nearly two weeks, mostly at the Dharavi slum. The climax portion was planned to be filmed at Madurai, which was considered as "apt" for the "action-oriented" sequence, but was eventually filmed in Chennai as well, while the remaining scenes were to be canned in Hyderabad. During the first week of June, Ajith had reportedly completed his portion, with his last day shoot being held in Hyderabad, while sources confirmed that filming was still being carried on later that month in Hyderabad. Shooting was further extended, with the crew leaving for Bangkok again in late June for a ten-day schedule to shoot the pending scenes, including a lengthy fight sequence and a song, involving Premji and Lakshmi Rai. Despite earlier announcement that Ajith had finished his portions, a "special scene" featuring Ajith in a different look was filmed on one day during the first week of July. Principal photography ended by June 2011.


title: Mankatha

text: "Mankatha"'s post-production works commenced by mid-June 2011, and were carried on for over one month. Earlier, an animation sequence lasting about 4 minutes was being created by specialised technicians, while CGI special effects were included in a song and action scenes, which was cited as the reason for the delay. By early August, all actors but Ajith had finished dubbing for their characters, including Trisha who on Venkat Prabhu insistence spoke dubbing for herself in the film, which became only the third film to feature her original voice. Rekhs, who had previously subtitled films including "Enthiran" and "Vinnaithaandi Varuvaayaa", subtitled "Mankatha" during the first week of August, while Yuvan Shankar Raja worked on the re-recording, being assisted by Premji.


title: Mankatha

text: The film score and soundtrack of "Mankatha" were composed by Yuvan Shankar Raja, becoming his fourth collaboration with Venkat Prabhu and Ajith as well. The soundtrack, consists of eight tracks, including one Theme music track and one club mix, with lyrics penned by Vaali, and Gangai Amaran and renowned poet Subramanya Bharathi's grandson, Niranjan Bharathi. As earlier done in "Saroja" and "Goa", a promotional track too was planned, which however did not materialise in last minute. Prior to the official soundtrack launch, a single track, "Vilaiyaadu Mankatha", was released in mid-May 2011. The music rights were secured by Sony Music who had reportedly offered 10 million. The soundtrack album, following several postponements, was released on 10 August 2011 at Radio Mirchi's Chennai station, while two days later the team arranged a press meet, showcasing two songs and the trailer of the film. The album was reported to have achieved record breaking sales. The songs received mixed response, with their placements in the film being criticised, while the film's score was widely appreciated.


title: Mankatha

text: Even before beginning the principal photography, a short teaser trailer was shot on the day of the launch itself, since it coincided with Aadi Perukku and was considered an auspicious day. The first teaser was screened along with the film "Naan Mahaan Alla", another Cloud Nine Movies distribution that released on 20 August 2010. During Diwali 2010, the first look posters of "Mankatha" were published in newspapers. A teaser featuring the song "Vilaiyaadu Mankatha" was released on Ajith's birthday, on 1 May 2011 on YouTube, after plans of releasing the film or the soundtrack on that day had failed. As the teaser garnered high response, the song was released as a single track on 20 May 2011, creating positive media response. "Mankatha" merchandise were launched for sale after the 50th day. Items included exclusive sun glasses, t-shirts, hand cuffs, and pendants etc.:- all on a limited edition basis.


title: Mankatha

text: "Mankatha" was given a U/A certificate by the Censor Board after 10 cuts. Prior to submitting the film for certification, the makers themselves bleeped profane words. One week later, the British Board of Film Classification issued a 15 certificate with an advice that it "contains strong threat and violence". Later, the distributor reduced violence in three reels and secured a 12A classification for theatrical release. During the late production stage, reports indicated that "Mankatha" faced "political pressure" following the change in government after the 2011 Tamil Nadu Legislative Assembly election, with sources claiming that the film did not find any buyers in Tamil Nadu, since it was produced by the grandson of DMK president M. Karunanidhi. Cloud Nine Movies began negotiations with other production houses to sell the domestical theatrical rights, however talks with UTV Motion Pictures and Gemini Film Circuit resulted in failure. On 22 August 2011, Gnanavelraja confirmed that his production house Studio Green, had purchased the Indian domestic theatrical and the television rights of the film at an undisclosed record amount.


title: Mankatha

text: However, in a turn of events, Studio Green cancelled the deal the very next day due to "various reasons" which Gnanavel Raja did not want to elaborate, in spite of posters in newspapers featuring Studio Green's logo. "Times of India" reported that Azhagiri bought back the rights, since Gnanavel Raja had planned to sell the satellite rights to Jaya TV, a channel run by the opposition party, AIADMK. On 24 August, Azhagiri announced that Kalanidhi Maran's Sun Pictures had bought the theatrical and satellite rights of the film and would distribute it along with Cloud Nine Movies. Udhayanidhi Stalin was said to have negotiated the deal and united the production houses to release the film jointly. "Mankatha" thus became the first Ajith film under Sun Pictures banner as well as their first release after the assembly election. Actress Raadhika's Radaan Mediaworks distributed the film to Tamil Nadu theatres.


title: Mankatha

text: Dhayanidhi Azhagiri announced that the film would be a worldwide release with subtitles in English, simultaneously opening across Singapore, Malaysia, Sri Lanka, United Kingdom, the United States, Australia, Canada and many other parts of Middle East and Europe. The film was also dubbed into Telugu as "Gambler" and released in Andhra Pradesh on 9 September 2011. Noted Telugu producer Bellamkonda Suresh acquired the film's Telugu dubbing rights by late August 2011 and released the Telugu dubbed version across 225 screens, enabling the biggest opening for a dubbed version of a Tamil film. Ten days later, the film was a box-office hit in Kerala as well. The film became scheduled for a release on 1 September 2011 in order to cash in on the Vinayaka Chaturthi-Ramadan weekend, before Ayngaran International eventually finalised 31 August 2011 as the release date in overseas theatres few days later. In the United States, the film was released at 34 theatres.


title: Mankatha

text: "Mankatha" released in about 1,000 screens worldwide and the film had a solo opening in Tamil Nadu on 31 August. It was said to have collected 25.2 crore nett from 370 screens in Tamil Nadu during the opening five-day weekend, and around 30 crore nett in its first week. The film became the biggest grosser of the year as well as that of Ajith's career, while also garnering the second-highest opening after "Enthiran" (2010). In Chennai city alone, the film earned 2.72 crore in the first weekend from 19 screens. The multiplexes gave it the maximum number of shows including morning shows in all screens. "Mankatha" recovered its cost of production in 2 weeks, making it the fastest film in recent times to turn profitable.


title: Mankatha

text: At the Mayajaal multiplex, "Mankatha" was screened in all 14 screens on the first day, resulting in 70 shows per day, all being sold out, while Sathyam Cinemas reported a net of 0.34 crore from two screens for the five-day weekend. The film grossed 6.5 crore in 19 days in Chennai. The Telugu version "Gambler", which released in 225 screens. In Kerala, the film was released in the original language in Thiruvananthapuram and Palakkad districts on 31 August while a dubbed version released all over the state on 9 September, opening at first rank, outclassing other Malayalam releases. The film bought for 0.6 crore in Kerala was expected to get distributor share of 1.4 crore. The Telugu version got high opening compared to other mainstream films It was successful at the Bangalore box office.


title: Mankatha

text: The film opened at second rank in Malaysia, grossing $803,666 in its first weekend, with a per screen average of $19,602 (highest per screen average) claiming the second-highest opening weekend for a Tamil film. After four weeks, the film grossed 0.6 crore in Malaysia. In the United Kingdom, the film grossed $179,054 from 16 screens, opening at No.1 spot and No.4 in the all-time chart Overall the film grossed $1,104,911 in Malaysia in six weeks, and $268,533 in UK at the end of the third week. Sun Pictures, the distributors declared that "Mankatha" grossed 80 crore worldwide in thirty days, nearing the end of its run. Sify termed the film as a blockbuster as well as the year's biggest commercial success. The film completed a 50-day run at the box office and was ranked as second highest grosser in Tamil cinema behind Enthiran at the time of its release. According to "The Times of India", the film grossed ₹100 crore at the worldwide box office in its final run. However, "Madras Musings" stated that the film made only 68 crore in its lifetime.


title: Mankatha

text: "Mankatha" received generally positive reviews with critics lauding Ajith's performance. Malathi Rangarajan of "The Hindu" called it a "gutsy" and "engaging game of cat and mouse that springs no surprises", adding that Venkat Prabhu had "laid out a filling spread this time". Anupama Subramanian of the "Deccan Chronicle" rated it 3 out of 5 as well, claiming that Ajith "sparkles in his 50th film". N Venkateswaran of "The Times of India" rated the film 4 out of 5, saying "Ajith is the soul of the movie and the others have nothing much to do, with the exception of Arjun." and called it "a good watch, especially because of Ajith's baddie act." S. Viswanath of "Deccan Herald" said, "Mankatha is strictly for Ajith fans, who has a wholesome blast, puffing, bulldozing his way and bedding belles by the dozen like there is no tomorrow." Pavithra Srinivasan of "Rediff" rated the film 2.5 out of 5 stars, saying "If Mankatha works even just a bit, it's because of Ajith, whose charisma shines through. Watch it only for him. The rest really don't matter." "Sify"s critic highlighted that Ajith "steals the thunder and plays the emotionless bad man, to perfection", further adding that he looked "smashing and his scorching screen presence is unmatchable", while the reviewer criticised the script as being "dull", and concluded " For die-hard fans of Ajith who don't have a problem with an unforgivable 2 Hours 40 minutes running time and juvenile comedy, this might be a treat. For others though, it's strictly average entertainment." Akhila Krishnamurthy of Outlook said, "The thing about testosterone is it can either excite or frustrate. There is no in-between. Tamil superstar Ajith's much-anticipated 50th film is a very "male film", no doubt. There's a heist, a few chases, gunfights, cusswords, three good-looking women and a salt-and-pepper-haired protagonist, who is naughty at forty. Only, none of it excites."


title: Mankatha

text: Google Zeitgeist 2011, a compilation of the year's most frequent search queries, placed "Mankatha" at 7th rank, becoming the only Tamil film to secure a place in the list. Sudhish Kamath of "The Hindu" included "Mankatha" in his list "Year of Anti-Hero" stating that "Mankatha earns its place here simply because it took a fairly dark genre like noir and celebrated evil minus the darkness" and also went to write that "this comic heist film is the best thing Ajith has ever done since Billa". Ajith's appearance, dubbed the "salt-and-pepper look", was widely praised, leading to him reusing the same appearance in most of his subsequent films. This look eventually became a trendsetter for Tamil heroes.


title: Mankatha

text: The verse "Aadama Jaichomada" from "Machi Open The Bottle" inspired a film of the same name. The scene where Vinayak gets excited by the collection of money was parodied in "Tamizh Padam 2" (2018).


title: Mankatha

text: Shortly after the film's release and its high commercial success, Venkat Prabhu confirmed the possibility of a sequel, provided that Ajith Kumar accepted. Sources also reported that Prabhu planned to use the initial script that he had penned before Ajith became part of the film. In November 2013, Dhayanidhi said, "That people are still talking about Mankatha 2 shows the grand victory that Mankatha achieved when it released. People can't forget such a movie and I want to make sure that 'Brand Mankatha' stays on in people's minds. The sequel might happen in future but we haven't started any discussions in this regard. The good news is that Venkat Prabhu has already readied the plot for the sequel.


title: Mankatha

text: A Hindi remake of the film has been confirmed by Gnanavel Raja of Studio Green. The cast and crew are yet to be finalised.


title: 2022 Italian presidential election

text: The 2022 Italian presidential election was held in Rome between 24 and 29 January 2022. The president of Italy was elected by a joint assembly composed of the Italian Parliament and regional representatives. The election process extended over multiple days, culminating in incumbent president Sergio Mattarella being confirmed for a second term, with a total of 759 votes on the eighth ballot. This was the second most votes (after Sandro Pertini in 1978) ever received by a presidential candidate. Mattarella became the second president to be re-elected, his predecessor Giorgio Napolitano being the first.


title: 2022 Italian presidential election

text: Mattarella had initially ruled out a second term. On 29 January, he agreed to serve a second term, as most party leaders and Mario Draghi, the prime minister of Italy, asked him to accept their joint nomination for another term. Mattarella had previously already received significant and growing support in several rounds of voting, namely 125 votes on the third ballot, 166 votes on the fourth ballot, 336 votes on the sixth ballot, and 387 votes on the seventh ballot. Among Italian presidential elections, the 2022 election had the highest number of ballots since 1992, when Oscar Luigi Scalfaro was elected on the 16th ballot.


title: 2022 Italian presidential election

text: The incumbent president Sergio Mattarella was elected in 2015 by the Italian Parliament and regional representatives. Mattarella was supported by the centre-left coalition, led by then-prime minister and secretary of the Democratic Party (PD) Matteo Renzi, and gained broad support. The result of the 2016 Italian constitutional referendum led to Renzi's resignation and to the formation of the Gentiloni Cabinet headed by Paolo Gentiloni, at the time a deputy from the PD and the incumbent Italian Minister of Foreign Affairs, as the prime minister.


title: 2022 Italian presidential election

text: The 2018 general election resulted in a hung parliament. As a result of the ensuing government formation talks, the Five Star Movement (M5S) and the League (Lega) formed a new cabinet led Giuseppe Conte, a law professor close to the M5S. During the government formation, Mattarella's role became particularly important when he refused to appoint the Eurosceptic economist Paolo Savona as minister of finance.


title: 2022 Italian presidential election

text: In August 2019, Lega's leader Matteo Salvini announced a motion of no confidence against Conte after growing tensions within the majority. Many political analysts believed that the no confidence motion was an attempt to force early elections and to improve Lega's standing in the parliament, so that Salvini could become the next prime minister. On 21 August, Mattarella started the consultations with all the parliamentary groups. On the same day, the national direction of the PD officially opened to form a new cabinet in a coalition with the M5S. On 29 August, Mattarella summoned Conte to the Quirinal Palace to give him the task of forming the new cabinet.


title: 2022 Italian presidential election

text: In 2020, Italy became one of the countries worst affected by the COVID-19 pandemic. Conte's government was the first in the Western world to implement a national pandemic lockdown to stop the spread of the disease. Despite being widely approved by public opinion, the COVID-19 pandemic lockdown in Italy was also described as the largest suppression of constitutional rights in the history of the Italian Republic. Mattarella described the COVID-19 crisis as "three emergencies on the health, social and economic fronts" that needed a stable government and made sure a new government could be formed after the Conte II Cabinet fell, amid high popularity, with the possibility of a second term.


title: 2022 Italian presidential election

text: In January 2021, Renzi's Italia Viva (IV), a member of the coalition supporting Conte's cabinet, withdrew its support for the government. Although Conte won confidence votes in the parliament in the subsequent days, he chose to resign, having failed to reach an absolute majority in the Senate of the Republic. After negotiations to form a third Conte cabinet failed, President Mattarella appointed Mario Draghi, the former president of the European Central Bank, as the prime minister at the head of a national unity government composed of independent technocrats and politicians from M5S, Lega, PD, IV, Forza Italia (FI), and Free and Equal (LeU).


title: 2022 Italian presidential election

text: During 2021, President Mattarella expressed his unavailability regarding a second term, which had been proposed by various political forces, recalling similar remarks made by his predecessors Antonio Segni and Giovanni Leone. Other political figures stated that they did not intend to run as president, among them Romano Prodi, Emma Bonino, and Liliana Segre.


title: 2022 Italian presidential election

text: During a press conference in Rome on 22 December 2021, Prime Minister Draghi, questioned on the hypothesis of his rise to the Quirinal Palace, declared that he was "a grandfather in the service of institutions". The declaration was interpreted as a confirmation of his willingness to be elected president and accelerated the debate around the consequences of a possible election to the presidency of the incumbent prime minister, something that had never happened before. Several national and foreign observers considered Draghi to be a probable candidate for the presidency, among them the "Corriere della Sera", "The Economist", "Le Figaro", the "Financial Times", "Le Monde", the "Süddeutsche Zeitung", and "The New York Times".