space mathematical concept space named notion space extends methods dimensional plane dimensional space spaces finite infinite number dimensions space abstract space possessing structure inner product allows length angle measured furthermore spaces complete limits space allow techniques spaces arise naturally frequently mathematics physics typically infinite dimensional function spaces earliest spaces studied point view decade 20th century tools theories partial differential quantum mechanics analysis includes applications signal processing heat transfer theory forms mathematical coined term space abstract concept diverse applications success space methods era functional analysis apart classical spaces examples spaces include spaces square functions spaces sequences spaces consisting functions hardy spaces functions geometric plays important role aspects space theory exact law hold space deeper level perpendicular projection analog dropping altitude triangle plays significant role problems aspects theory element space uniquely specified coordinates respect set coordinate axes basis analogy coordinates plane set axes infinite means space thought terms space infinite sequences square space older literature referred space linear operators space likewise fairly concrete objects good cases simply stretch space different factors mutually perpendicular directions sense precise study spectrum definition illustration example space familiar examples space space consisting dimensional denoted equipped dot product dot product takes produces real number represented coordinates dot product defined formula dot product properties linear argument positive definite equality operation pairs like dot product properties known real inner product space equipped inner product known real inner product space finite dimensional inner product space space basic feature dot product connects geometry related length norm denoted angle means formula formula space relies ability limits useful criteria concluding limits exist mathematical series formula consisting absolutely provided sum lengths ordinary series real numbers formula series series absolutely limit space sense formula property expresses space series absolutely ordinary sense spaces taken complex numbers complex plane denoted equipped notion magnitude complex defined square root product complex formula real imaginary parts usual dimensional length formula inner product pair complex numbers product complex formula complex valued real gives usual dimensional dot product second example space elements pairs complex numbers inner product given formula real dimensional dot product inner product means result complex formula definition space real complex inner product space complete metric space respect distance function induced inner product complex inner product space means complex space inner product formula complex number pair elements following properties inner product pair elements equal complex inner product swapped elements formula inner product linear argument complex numbers formula inner product element positive definite formula case equality holds precisely follows properties complex inner product second argument meaning formula real inner product space defined way real space inner product takes real values inner product linear argument norm real valued function formula distance points defined terms norm formula function distance function means distance zero distance positive triangle inequality holds meaning length leg triangle exceed sum lengths legs formula property ultimately consequence fundamental inequality asserts formula equality dependent relative distance function defined way inner product space metric space known pre space space additionally complete space space expressed form criterion sequences space complete sequence respect norm element space characterized following equivalent condition series formula absolutely sense formula series sense partial sums converge element complete space spaces definition spaces spaces notions like defined special importance notion closed linear space inner product induced restriction complete closed set complete metric space space right second example sequence spaces sequence space consists infinite sequences complex numbers series formula inner product defined formula series consequence inequality space holds provided series elements absolutely norm element proof basic mathematical analysis permits mathematical series elements space manipulated ease series complex numbers finite dimensional space history prior development spaces spaces known particular idea abstract linear space gained traction end 19th century space elements added multiplied real complex numbers necessarily identifying elements geometric position momentum physical systems objects studied turn 20th century particular spaces sequences including series spaces functions naturally thought linear spaces functions instance added multiplied constant operations obey laws satisfied addition spatial decade 20th century parallel developments led introduction spaces observation arose study integral square real valued functions interval inner product formula familiar properties dot product particular idea family functions meaning exploited similarity inner product usual dot product prove analog operator form formula continuous function resulting expansion expresses function series form formula functions sense individual terms series referred elementary product solutions expansions fail converge suitable sense square function missing ingredient ensures convergence second development integral alternative integral introduced integral possible integrate broader class functions fischer independently proved space square functions complete metric space consequence geometry 19th century results series easily carried general spaces resulting apparatus usually known fischer basic results proved early 20th century example representation independently established coined term abstract space work operators studied particular spaces great detail physically motivated point view gave complete treatment later seminal work foundations quantum mechanics continued work space soon adopted example book quantum mechanics theory groups significance concept space realization offers best mathematical quantum mechanics short states quantum mechanical system certain space operators space system operators measurements projections relation quantum mechanical operators provided impetus development representation theory groups initiated work hand early 1930s clear classical mechanics described terms space von classical mechanics certain properties classical systems analyzed space techniques framework theory quantum mechanics naturally operators defined space according matrix mechanics quantum theory began investigating operator 1930s rings operators space kind studied contemporaries known von 1940s gave definition kind operator called hand reference underlying space useful features operator previously studied self operators particular existing space theory techniques basic abstract analysis representation theory examples spaces spaces function spaces associated measure spaces set measure let space complex valued functions integral square absolute value function finite function formula functions identified differ set measure zero inner product functions defined formula integral exists inequality defines inner product space equipped inner product fact complete integral essential ensure domains real numbers instance functions spaces appear natural settings spaces square functions respect measure real line unit interval respectively natural domains define transform series situations measure ordinary measure real line instance positive function space functions interval satisfying formula called weighted space called weight function inner product defined formula weighted space identical space measure set defined formula weighted spaces like frequently study different families respect different functions spaces spaces denoted spaces special kind function space performed unlike spaces spaces support structure inner product permitted spaces convenient setting theory partial differential form basis theory direct methods variations non negative space contains functions weak derivatives order inner product formula dot indicates dot product space partial derivatives order spaces defined spaces studied point view theory relying specifically space structure suitable domain define space space roughly formula understood terms mapping apart providing definition spaces non definition particularly desirable properties transform ideal study operators methods compact obtain instance basis theory spaces functions hardy spaces hardy spaces function spaces arising complex analysis analysis elements certain functions complex domain let denote unit disc complex plane hardy space defined space functions means formula remain bounded norm hardy space defined formula hardy spaces disc related series function formula formula consists functions circle negative frequency spaces spaces family spaces functions let bounded open set complex plane higher dimensional complex space let space functions sense formula integral taken respect measure clearly fact closed space right consequence estimate valid compact formula turn follows integral formula convergence sequence functions implies compact convergence limit function consequence inequality linear functional function point actually continuous representation implies evaluation functional represented element function formula formula known integral property formula space example space space functions property analogous hardy space admits known common areas mathematics instance analysis space square functions unit ball space consequence mean value functions applications applications spaces exploit fact spaces support simple geometric concepts like projection change basis usual finite dimensional setting particular theory continuous self linear operators space usual matrix plays major role applications theory areas mathematics physics theory theory ordinary differential methods suitable space study behavior differential example problem arises study waves violin string drum central problem ordinary differential problem differential equation form formula unknown function interval satisfying general robin boundary conditions formula functions given advance problem find function equation solution problem solutions certain values called system consequence compact operators applied integral operator defined green function system furthermore consequence general result system arranged increasing sequence tending infinity partial differential spaces form basic tool study partial differential classes partial differential linear possible consider solution known weak solution class functions weak involve class functions space suitable weak reduces problem problem finding solution important showing solution exists unique given boundary data linear result ensures unique large class problems strategy forms method finite element method numerical solution partial differential typical example equation boundary conditions bounded domain weak consists finding function continuously functions boundary formula recast terms space consisting functions weak partial derivatives square boundary question reduces finding space space formula continuous form continuous linear functional given respectively formula equation follows inequality form ensures existence solutions equation spaces allow partial differential formulated similar way basic tool analysis suitable modifications similar techniques applied partial differential certain partial differential theory field theory study long term behavior chaotic systems case field theory applies microscopic state system extremely complicated impossible understand ensemble individual collisions particles matter average behavior sufficiently long time intervals laws average behavior particular law asserts sufficiently long independent measurement system total energy form temperature system apart energy measured independent quantities phase space explicitly suppose energy fixed let phase space consisting states energy energy surface let denote evolution operator phase space system continuous non constant functions formula time implies exists measure energy surface time translation result time translation transformation space consisting square functions energy surface respect inner product formula von mean states following strongly continuous operators space projection space common fixed points formula system fixed set time evolution consists constant functions implies following function formula long time average equal expectation value energy surface analysis basic goals analysis function possibly infinite linear combination given basis functions associated series classical series associated function defined interval series form formula formula example adding terms series function shown figure basis functions waves shorter fundamental wave basis functions fundamental additional summed terms called phenomenon significant problem classical series asks sense series function space methods provide possible answer question functions form basis space consequently square function expressed series formula series space sense mean problem studied abstract point view space basis element space written unique way sum basis elements appearing basis elements known element space especially useful natural use different basis functions space circumstances desirable function functions instance higher dimensions spherical instance basis functions given function finite linear combination formula selected magnitude difference small possible best projection consisting linear combinations calculated formula formula difference consequence inequality formula applications physical problems function physically meaningful differential operator typically operator forms foundation study functions reference spectrum differential operator concrete physical application involves problem hearing shape drum given fundamental modes vibration capable producing shape drum mathematical question involves equation plane represent fundamental modes vibration direct analogy represent fundamental modes vibration violin string theory certain aspects transform function analysis function defined compact set spectrum corresponds vibrations violin string drum transform function function defined space components continuous spectrum transformation sense precise asserts space time domain frequency domain property transformation recurring theme abstract analysis evidenced instance spherical functions occurring analysis quantum mechanics rigorous quantum mechanics developed possible states precisely pure states quantum mechanical system represented unit called state residing complex space known state space defined complex number norm phase factor words possible states points space usually called complex space exact nature space dependent system example position momentum states single non spin zero particle space square functions states spin single unit elements dimensional complex space represented self linear operator acting state space corresponds operator associated corresponds value inner product state complex number known probability ideal measurement quantum mechanical system probability system collapses given initial state particular given square absolute value probability initial final states possible results measurement operator explains choice self operators real probability distribution given state found computing corresponding operator general system states typically pure instead represented statistical pure states mixed states given density self operators trace space general quantum mechanical systems effects single measurement influence parts system manner described instead positive operator valued measure structure states general theory considerably complicated pure states color perception true physical color represented combination pure colors physical colors composed number physical colors space physical colors represented space colors humans types cone cells color perception human colors represented dimensional space non unique linear mapping space physical colors space human colors explains distinct physical colors perceived humans identical pure yellow light versus mix red green light properties identity space formula generally means element formula induction extended family formula identity stated valid inner product space required extension identity series series series squares norms formula furthermore sum series independent order taken identity definition space space furthermore space following identity holds formula space identity holds space inner product uniquely determined norm identity real spaces identity formula complex spaces formula law implies space uniformly convex space best employs projection non closed convex space point exists unique point distance points formula equivalent saying point minimal norm translated convex set proof consists showing sequence identity point minimal norm generally holds uniformly convex space result applied closed shown point closest characterized formula point projection mapping linear projections result especially significant applied mathematics especially numerical analysis forms basis squares methods particular equal find non zero select useful criterion obtained applying observation closed generated spans dense sole dual space space continuous linear functions space base field carries natural norm defined formula norm law dual space inner product space dual space complete space right representation convenient description dual element unique element defined formula mapping formula mapping representation states mapping element dual exists formula inner product dual space formula reversal order right hand real case dual actually real spaces naturally representing obtained following way closed equal exists non zero suitable multiple requirement yields formula correspondence exploited bra popular physics common physics assume inner product denoted linear right formula result seen action linear functional bra representation relies fundamentally presence inner product space fact implies dual inner product space identified completion immediate consequence representation space meaning natural map double dual space weakly sequences space sequence weakly formula example sequence weakly consequence inequality weakly sequence bounded uniform principle conversely bounded sequence space admits weakly fact prove results continuous convex way continuous functions variants simple statement follows convex continuous function tends tends admits minimum point fact fundamental direct methods variations results convex direct consequence slightly abstract fact closed bounded convex space weakly compact existence weakly special case space properties general property spaces continues hold spaces open mapping states continuous linear transformation space open mapping meaning sends open sets open sets bounded continuous linear function space continuous linear map continuous considerably simpler prove case spaces general spaces open mapping equivalent closed asserts function space continuous closed set case spaces basic study operators closed operator asserts closed convex set separated point outside means space immediate consequence best property element closed convex set closest separating plane perpendicular segment passing operators spaces bounded operators continuous linear operators space second space bounded sense map bounded sets bounded sets conversely operator bounded continuous space bounded linear operators norm operator norm given formula sum composite bounded linear operators bounded linear map sends linear continuous according representation represented form formula defines bounded linear operator set bounded linear operators addition composition operations norm operation type operator element called self called non negative written equality holds called positive set self operators admits partial order form non negative positive true sense non negative operator exists unique non negative square root formula sense precise self operators thought operators real element called normal normal operators sum self operators imaginary multiple self operator formula normal operators thought terms real imaginary parts element called given expressed requiring operators form group composition group element compact sends bounded sets relatively compact sets bounded operator compact bounded sequence sequence integral operators compact fact define special class operators known operators especially important study integral operators differ compact operator multiple identity characterized operators finite dimensional index operator defined formula index plays deep role differential geometry singer index operators operators spaces important applications quantum mechanics operator space defined linear operator domain linear domain dense case known densely defined operator densely defined operator defined essentially manner bounded operators self operators play role mathematical quantum mechanics examples self operators space suitable extension differential operator formula imaginary unit function compact support operator formula correspond momentum position respectively note defined case derivative need exist case product function need square cases set possible arguments form dense constructions direct sums spaces combined space called direct sum denoted formula consisting set ordered pairs inner product defined formula generally family spaces direct sum denoted formula consists set families formula product formula inner product defined formula included closed direct sum conversely system closed space union dense direct sum case called internal direct sum direct sum internal external equipped family projections direct projections bounded self operators satisfy condition formula compact self operators space states splits direct sum operator gives explicit operator sum projections direct sum spaces appears quantum mechanics space system containing variable number particles space direct sum corresponds additional degree freedom quantum mechanical system representation theory peter guarantees representation compact group space splits direct sum finite dimensional representations products defines inner product ordinary product follows simple let formula formula extends inner product product denoted formula space obtained completing metric associated inner product example provided space product copies space square functions square sends simple function formula square example typical following sense associated simple product rank operator maps given formula formula mapping defined simple extends linear identification space finite rank operators extends linear product formula space operators bases notion basis linear case spaces space basis family elements satisfying conditions different elements formula element family norm linear span family dense system satisfying conditions basis called system set sequence system independent system space related fact dense linear zero set closure linear span space examples bases include set forms basis dot product sequence forms basis complex space infinite dimensional case basis basis sense linear distinguish basis called basis span basis dense implies space written sum infinite series implies unique sequence spaces space square sequences complex numbers set infinite sequences formula complex numbers formula space basis formula generally set form space sequences index set defined formula defined formula taken finite follows sum finite element terms space space inner product formula sum terms inequality basis set given formula inequality formula let finite system arbitrary let formula follows identity twice follows formula let arbitrary system applying preceding inequality finite gives inequality formula according definition sum arbitrary family non negative real numbers inequality implies projection linear spanned norm exceed dimensions assertion length leg right triangle exceed length inequality stepping stone powerful identity case inequality actually equality basis element written formula inequality guarantees expression defined consists terms sum called expansion individual formula formula conversely set identity holds basis dimension consequence space admits basis furthermore bases space called dimension space instance basis dimension finite cardinal number consequence identity basis map defined spaces linear mapping formula cardinal number dimension space sequence space set spaces space admits basis infinite dimensional spaces past spaces required definition spaces physics refers infinite dimensional space space space quantum field theory spaces fact stipulated argued non spaces important quantum field theory roughly systems theory possess infinite number degrees freedom infinite product spaces dimension greater non instance field naturally thought element product factors represent point space perspective natural state space non space small product contain physically meaningful fields defined non space models state infinite collection particles region space basis space density particles continuous set possible basis projections space set defined formula closed proved easily continuity inner product forms space closed called complement fact written uniquely internal direct sum linear operator maps called projection natural correspondence set closed set bounded self operators specifically projection self linear operator norm property self linear operator form range unique element distance provides interpretation best elements called mutually equivalent sum projections projection case composite generally projection fact composite projection projections case restricting space projection gives rise projection mapping inclusion mapping formula meaning formula operator norm projection non zero closed equal formula closed space image operator norm property possessing appropriate projection operators characterizes spaces space dimension higher space closed operator norm image formula result characterizes metric structure space structure space space characterized terms presence space space closed closed equal internal direct sum complement elementary results function sense formula equality holding contained closure result special case closure completely characterized terms complement closure equal formula complement connection partial order space general complement sum intersection formula addition closed formula theory developed theory self operators space roughly analogous study self complex numbers sense obtain self operator suitable sum actually integral projection operators spectrum operator denoted set complex numbers lacks continuous bounded spectrum compact set complex plane lies inside disc formula self spectrum real fact contained interval formula actually contained spectrum operator given formula unlike finite element spectrum linear operator lack elements spectrum operator general sense known values values need subtle finite dimensions self operator takes particularly simple form addition assumed compact operator compact self operators states compact self operator values spectrum limit point complex plane possibly zero direct sum formula denotes projection formula sum respect norm plays fundamental role theory integral integral operators compact particular arise operators general self operators involves kind operator valued integral infinite family associated associates real number operator projection operator positive self operator defined formula operators increasing relative partial order defined self operators correspond precisely jump asserts formula integral understood integral respect norm particular ordinary valued integral representation formula somewhat similar holds normal operators spectrum contain non real complex numbers operator valued measure instead replaced resolution identity major application methods mapping allows apply self operator continuous complex function defined spectrum forming integral formula resulting continuous functional applications particular operators theory self operators marginally difficult bounded operators spectrum operator defined precisely way bounded operators value operator formula fails defined continuous operator self guarantees spectrum real essential idea working operators look instead non real bounded normal operator admits representation transferred representation similar strategy instance study spectrum operator address operator directly instead looks associated potential potential precise version case given densely defined self operator space corresponds unique resolution identity sets formula measure concentrated spectrum version applies normal operators
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