Welcome to lecture 35 on measure and integration. In the previous lectures, we had been looking at the pth power integrable functions on a measure space xs mu and we have studied some general properties of these function spaces. Today, we will look at in slightly more in detail the special subspace, namely when p is equal to 2. So, we will look at today. The topic for today's discussion would be the space L2 xs mu, the space of complex valued square integrable functions. This is a special space in the sense that this can be viewed as a generalization of the Euclidean spaces. Recall, in Rn, the Euclidean space, we have the notion of dot product of vectors, which is related to the magnitude of the vectors and also help us to define the notion of angles on Rn. Let us see what we can do as far as L2 is concerned. We look at the space L2 of xs mu to be the space of all functions, which are defined on x, which are complex valued such that mod of f square d mu is finite. For such spaces for a function f in L2 of x, we have defined the norm of this, which is equal to integral of mod f square d mu raise to power 1 by 2. We said that this is very much like the magnitude in Rn. So, this is similar to magnitude in Rn. Let us just briefly recall what was the magnitude in Rn. For a vector say x in Rn, the Euclidean norm is defined as look at the components. If x is having components x1, xn, then look at mod of xi square sigma 1 to n and the square root of that. For a function f in L2, let us regard f as a vector with fx as the component. This is the xth component of f. How would we define the norm? Look at the xth component fx. Look at square. This is very much similar to what we have done for Euclidean. Look at the component square. Now, sum it up, but here the summation is over x and x is any x belonging to x, any indexing set. So, it is nothing but integral d mu and then the power 1 by 2. In that sense, this is a perfect generalization of the Euclidean norm to arbitrary spaces. Now, on L2, we have the notion of for a vector x belonging to Rn with components xn. We have the notion of what is called the dot product. So, dot product if x and y are two vectors with y components as yn, then x comma y, the dot product is defined as xi yi, i equal to 1 to n. If we are in a complex plane, so if x and y belong to cn, then the dot product xy is defined as sigma xi yi bar i equal to 1 to n. This notion of dot product is related to the magnitude in the following way that in either Rn or cn, either one, the norm of x square is equal to the dot product of x with itself. We know that the dot product in Rn or cn gives a notion of angle and orthogonality, which helps us to do geometry in Rn. So, the basic idea of today's lecture would be on the space L2 of x, we already have the notion of norm, the notion of distance. We will define the notion of inner product or the notion of dot product on L2 of x and show how it is related to the notion of distance. So, that helps us to define the notion of orthogonality or perpendicularity of two elements in L2. So, we can do geometry in L2 of x as mu. So, let us define what is the notion of the dot product in L2. So, for functions f and g in L2, keep in mind our spaces are complex valued. So, define the dot product of f with g or also called the inner product of f with g as integral of fx g bar x d mu x, where this g bar is the complex conjugate of the function g. So, g bar of x is gx bar. So, this is defined, so the inner product or the dot product between f and g for two functions f and g is defined as the integral of fx gx bar d mu x, which is perfectly similar to in complex c to the power n a i b i bar. So, the first thing we want to say that this is well defined and that follows from the holders inequality that we had proved. Recall holders inequality that said that if f is a function in Lp and g is a function in Lq, then f into g is a function which is integrable and the integral of fg is less than or equal to the L2 norm of. So, here is a typo. So, let me just say it once again what we are meaning here, that should be integral. So, for holder inequality we had that if f belongs to Lp and g belongs to Lq, then fg belongs to L1 and integral mod fg integral of mod of f into g d mu is less than or equal to the pth norm of f and the qth norm of g. So, for p equal to 2, so that will give us that integral of fg d mu and p is equal to 2, q is equal to 2. So, we have got that 1 over p plus 1 over q is equal to 1. So, the holder inequality will give us that this is less than or equal to L2 norm of f into L2 norm of g and this is nothing but our, so this is nothing but this is bigger than or equal to fg inner product is less than or equal to this. So, that gives us the Cauchy Schwarz inequality. So, this is also called the Cauchy Schwarz inequality namely the inner absolute value of the inner product between f and g is less than or equal to the norm of f into norm of g. So, that says that this inner product is well defined, it is a well defined quantity. So, for every f and g in L2, we have the notion of the inner product. So, f into f comma g which is the inner product is well defined quantity and this behaves perfectly similar to that of the inner product for the ordinary vectors in R n or C n. That means, this is a function which is defined on L2 cross L2 and has the following properties namely the inner product of f with f is always bigger than or equal to 0 and the equality holds if and only if f is equal to 0. So, let us look at this property that how is this true?  So, this is always bigger than or equal to 0 and this will be equal to 0 if and only if our function mod f is equal to 0 almost everywhere. So, if and only if f is belonging to L2 as a treated as an element of L2, f is equal to 0. So, the first property is obvious namely the dot product of f with itself is always bigger than or equal to 0 and the second property says that the dot product or the inner product of f with g is same as the inner product of g with f. So, we are interchanging f and g and the complex conjugate of it. So, the inner product of f with g is same as the complex conjugate of the inner product of g with f and that is quite simple to verify from the definition. So, if we have got the inner product of f with g, so the inner product of f with g is equal to f g bar d mu and that is equal to, so inner product of f with g is equal to f g bar and that is equal to f bar g integral d mu bar. So, that is equal to and this is equal to g comma f bar. So, that verifies the property namely the inner product of f with g is equal to the inner product of g with f bar. Similarly, it is easy to verify using that the integral is linear, it is easy to verify that the inner product is linear in the first variable that means, alpha f plus beta g inner product with h is equal to alpha times the inner product of f with h plus beta times the inner product of g with h. Similarly, in the second variable this is conjugate linear because of the property of 2. So, f inner product with alpha g plus beta h is same as alpha bar of f g inner product f g plus beta bar of inner product of f h. So, in the second variable it is not linear it is a conjugate linear and finally, this property is obvious namely the L 2 norm of f is square root of the inner product of f with itself. So, all the properties that we have for the dot product in R n or C n are defined for the inner product in L 2. This is not very special for L 2. In fact, one can look at any vector space h over the field of real or complex numbers and if one has a function which is defined on h cross h taking values in the underlying field of real or complex having properties similar to that 1, 2, 3, 4, 5 of L 2. So, one can define what is called a inner product space. So, in general a inner product space is defined to be a vector space h on which there is a notion of inner product defined and what is a inner product it is a function defined on h cross h to R with those properties. So, once we have one has a notion of inner product that gives rise to notion of magnitude by the property that the magnitude of a vector u in h is nothing but is defined as the dot product of u with itself square root. And one verifies that Cauchy Schwarz inequality holds for this kind of inner product and that means this is a well defined norm. So, Cauchy Schwarz inequality will say that is a norm defined on it and once you have the notion of norm that gives rise to a metric on the underlying vector space and one can ask whether it is complete under that metric or not. So, if on a vector space inner product is defined. So, it becomes a inner product space and inner product space gives rise to a norm and if the underlying metric induced by the norm is complete one says h is a Hilbert space. So, that is the general definition of a Hilbert space. So, our L 2 is an example of a Hilbert space because L 2 x s mu is a vector space on which a notion of norm the L 2 norm is defined and that L 2 norm is related to the inner product just now we have seen and we have already seen as Rie's Fisher theorem which said that L 2 x s mu is a complete metric space in the L 2 metric. So, this is an example of a Hilbert space. Now, for once we have the notion of the inner product one can define the notion of two elements in L 2 to be orthogonal or perpendicular to each other. So, we say two elements f and g in L 2 are orthogonal to each other if the inner product between them is equal to 0 and that is what we have for vectors in R n also that the dot product is equal to 0. So, and we write this as f perpendicular to g. So, f perpendicular to g is defined as saying that the inner product of f with g is equal to 0. Now, we can also define the inner product f element f orthogonal to a subset s. So, writing it as f orthogonal to a subset s means that f is perpendicular to every element of s. So, f is perpendicular to s will mean that f is f comma h the inner product is equal to 0 for every element h of h in s. So, similarly we can define orthogonality of two sets also. With this one can prove what is called the Pythagoras identity namely if f and g are two functions in L 2 and f is orthogonal to g then the norm square of f plus g is equal to norm f square plus norm g square. So, let us just quickly verify the Pythagoras identity namely if f and g are two elements in L 2 and f is orthogonal to g then we want to compute the L 2 norm of square of this. By definition this is related to the inner product.   So, these two terms the inner product of g with f and f with g both are equal to 0. So, we get what is called the Pythagoras identity namely the norm of f plus g square is equal to norm f square plus norm g square whenever f is orthogonal to g. Let us carry over this idea of orthogonality a bit further. So, let us take as any non-empty subset of L 2. We call as a subspace of L 2 those who have done a bit of linear algebra will recognize this definition L 2 is a vector space. So, we are looking at a vector subspace of L 2. So, a set S is a non-empty subset is called a subspace of L 2 if for any f and g in f and g in S. So, this should be in S not in L 2 and alpha beta in complex number alpha f plus beta g belongs to S then we say S is a subspace. So, this is not L 2 that is S. So, that means for any two elements alpha beta in S the linear combination alpha f plus beta g should be in S in that case it is called a subspace of S and a subspace of S is called a closed subspace if it is closed under the metric on L 2 that is L 2 metric. So, it should be a closed set that means what that means whenever we have got a sequence f n in S and f n converges to a function f in L 2 norm then the limit must also be inside S. So, in that is what is the definition of a closed source subspace it is a subspace and it is a closed set under the L 2 metric. So, that is the notion of a closed subspace for given a set S in L 2 will denote by S upper suffix perpendicular. This is also called the orthogonal complement of S to be all elements in the space L 2 which are perpendicular to all elements of S. So, given a set S we are looking at all elements in L 2 which are orthogonal to every element of S. So, that is called S perpendicular and this is called the orthogonal complement of S and the claim is that this is a closed subspace of L 2. So, let us verify this fact that S a subset of L 2 and S perpendicular is the set of all elements f in L 2 such that f perpendicular to H for every H in S. Then claim is that first of all S perpendicular is a subspace S perpendicular is a subspace. So, let us take some H and G belonging to S perpendicular and alpha and beta belonging to C. Then alpha H plus beta G comma let us take an element f for f in S perpendicular for F in S. This will be equal to alpha times H f plus beta times G f using the property of linearity in the first variable for the inner product. Now, because f belongs to S and H and G are in S perpendicular, so this quantity inner product is 0 and the second inner product is 0. So, this sum this inner product is equal to 0. So, that says if H and G belong to S perpendicular and alpha and beta are in C, then alpha H plus beta G is always orthogonal to every element of S. Hence, it belongs to S perpendicular. So, S perpendicular is a subspace of. The next, let us prove that this is a closed subspace. So, next we want to check that it is a closed subspace. So, let f n belong to S perpendicular and f n converge to f in L 2. We want to check, so claim that f belongs to S perpendicular. So, for that, let us take any element H belonging to S and compute, we want to compute f comma H and the claim is this f comma H. So, this is equal to f comma limit n going to. So, what is H? This is not true. Let H belong to S. Now, the claim is that since f n converges to f in L, converges to f in L 2. So, this is equal to limit n going to infinity of f n comma H. So, this is a very simple thing to verify, because if we look at the difference of the two. So, if we look at f H, so why is this true? This is true because we look at this inner product of f with H and inner product of f n with H and look at the absolute value of this. Then we can write this as, so this quantity is equal to absolute value of f minus f n inner product with H and with this by Cauchy-Schwarz inequality, this is less than or equal to L 2 norm of f minus f n and L 2 norm of H. This goes to 0. So, we get therefore, f with H inner product is equal to limit n going to infinity inner product of f n with H. Since each f n belongs to S perpendicular, H is in S. So, that implies that each term is equal to 0. So, this is equal to 0 for every H belonging to S. So, that implies that f belongs to S perpendicular. So, this proves that S perpendicular for any set S, if we look at its orthogonal complement, then that is a close of space of H. So, this is also called the orthogonal complement of set. We next state an important result which seems geometrically obvious, which can be proved for any Hilbert space. So, we will just look at it for our space L 2. We will not prove this result. We will just assume this result and the proof can be referred to in the book. So, the result says that if f is a L 2 function and S is a close subspace of L 2, then look at the number alpha, which is an infremum of all the distances, L 2 distances of f from G, where G is any element in S. Then, the theorem says that this infremum is attained at some point in S. So, that means there exists not only it is attained, then there is a unique function f 0 belonging to S such that this infremum alpha is equal to norm of f minus f 0. Further, if this f does not belong to S, then look at the difference of f minus f 0, that is always going to be perpendicular to S. So, that is the claim of the theorem. We will not prove it, we will just geometrically analyze this result a bit. So, here is a close subspace S. So, look at the close subspace S of L 2. This is a close subspace of L 2 and we have got a function f, which is outside it. So, what we are going to do is, we are going to look at any point inside S, A point G and look at the L 2 distance of this. So, look at the L 2 distances of various points. So, it says that there is a point which is there is a value called f 0 such that this distance L 2 distance of f from it is the minimum. So, this is 0. So, look at f minus f 0 and this says if I look at f minus f 0, that is going to be orthogonal to it. So, that is the theorem. So, there exists a unique point f 0 belonging to S such that alpha the infremum is equal to the distance of f minus f 0 and f minus f 0 is perpendicular to S. So, look at this vector f minus f 0, that is always orthogonal to this S. So, geometrically in a sense, given a point and given a subspace, it is kind of the projection. The projection gives you the minimum. So, this is the generalization of the projection theorem for finite dimensional spaces. So, this is also called the best approximation theorem for Hilbert spaces. So, let us just say look at once again. It says that if you are given a close subspace of L 2 and look at and you are given a function f in L 2, look at alpha the minimum of the distances between f and elements of G says there is a value. There is a function in S which where this value is attained and as a consequence of this, it also says that if S is the proper close subspace of L 2, then its perpendicular cannot be 0 because there is a if it is a proper, then there is an element f minus f 0, which is not 0, which is orthogonal to it. So, as a immediate consequence that if S is a proper subspace proper close subspace of L 2, then its orthogonal complement cannot be 0. It has to something else. So, that also means that if the orthogonal complement of something is 0, then S must be equal to L 2. Another way of stating the same thing is this. So, as I said, we will not be proving this theorem, but we will give some applications of this today. So, let us look at some properties of orthogonal complement before we go on to prove some general facts. So, let us take S 1 and S 2 be subsets of L 2, just subsets, then the following properties hold namely S 1 perpendicular is a close subspace that we have already shown and S 1 perpendicular and S 1 they intersect only at the most at 0. They are just sets S 1 perpendicular is a subspace, because S 1 may not be a subspace. So, it says if S intersection S perpendicular is always inside 0 and that is obvious, because if f belongs to S 1 intersection S 1 perpendicular, then that means the inner product of f with itself, because f belongs to S 1 and it also belongs to S perpendicular that must be equal to 0. So, that implies norm of f is equal to 0 and that implies f must be equal to 0. So, it says so hence S 1 intersection S 1 perpendicular is inside 0. So, as obvious consequence, if S 1 is also a subspace, then S 1 intersection S 1 perpendicular, they do not have anything common other than the vector. The second property says that if S 2 is a subset of S 1, then S 1 perpendicular is a subset of S 2 perpendicular and that is obvious, because if we take any element say H in S 1 perpendicular, then H the dot product or inner product of H with every element of S 1 is equal to 0 and in particular with S 2 is equal to 0. So, that also belongs to S 2. So, this property is obvious. The third property says that S 1 is a subset of S 1 perpendicular. So, orthogonal complement of the orthogonal complement always includes S 1. So, that property is again obvious, because if we take f belonging to S 1 and H belonging to S 1 perpendicular, then we know that. So, then it implies H comma f the dot product is equal to 0, because f belongs to S 1 and H belongs to S 1 perpendicular and that is equal to 0, but that means H is perpendicular to f and so that means, so this means that for every H in S perpendicular f comma H or H comma f is equal to 0, that means f is belonging to S 1 perpendicular perpendicular. So, S 1 is always a subset of S 1 perpendicular perpendicular. We want to show that, so in case S 1 perpendicular perpendicular equal to S 1, in case these two are equal, then the left hand side is an orthogonal complement of a subspace. So, this is a close subspace. So, this is a close subspace implies S 1 is a close subspace. So, implies S 1 is a close subspace and let us prove the converse part, namely the converse is also true. So, suppose S 1 is equal to S 1 perpendicular perpendicular, then the claim that S 1 is a subspace, S 1 is a close, then S 1 is a close subspace. So, that we have already shown, we want to prove the other way round, so this is not what we want to is. So, suppose S 1 is a close subspace, so converse is if S 1 is a close subspace, then we want to show S 1 is also equal to S 1 perpendicular perpendicular. So, to prove this, suppose not.  So, let us assume there is a S 1 perpendicular perpendicular f naught in S 1. So, let us assume that is so. So, in that case, let us apply our best approximation theorem. So, implies by the theorem, just now we stated which we did not prove that there exist an element f naught belonging to S 1 such that f minus f naught is perpendicular to S 1. So, that means f minus f naught belongs to S 1 perpendicular, but also we have got f minus f naught belongs to, so there is a element f is in not in S 1, so there is element in f naught in S 1 such that the difference is perpendicular to S 1. Now, let us observe that S 1, now let us observe that this element f naught belongs to S 1, f naught belongs to S 1, which is contained in S 1 perpendicular, perpendicular. So, S 1 which is belonging to S 1, so that implies, so we have got f belonging to S 1 perpendicular perpendicular and f naught also belonging to, that means f naught S naught, f minus f naught belongs to S 1 perpendicular perpendicular and f naught is also in the same thing and this being a subspace, the difference must also belong to, this belongs to S 1 perpendicular perpendicular. Now, the element f minus f naught belongs to S 1 perpendicular perpendicular and it also belongs to S 1 perpendicular. So, from here and here it follows, so it belongs to a subspace and orthogonal complement of it, that means f minus f naught must be equal to 0, implying f is equal to f naught. So, that means what, that means f belongs to f naught f and where is f naught, f naught is in S 1, so this f also belong, so we would start with an f in S 1 perpendicular perpendicular and we are getting that f is equal to S naught, where S naught in element is S 1, so that implies that f belongs to S 1. So, what we have shown is, whenever f belongs to S 1 perpendicular perpendicular, it also belongs to S 1, so these two are equal. So, that proves the fact that if S 1, that proves the fact that if S 1 is equal to S 1 perpendicular perpendicular, then S 1 is a closed subspace of it. And next, let us observe the fact that if S 1 and S 2 are two closed subspaces and S 1 is perpendicular to S 2, then S 1 plus S 2 is also a closed subspace. So, let us observe that S 1, S 2 are closed subspaces and S 1 is perpendicular to S 2. So, let us take two elements, so let us look at S 1 plus S 2.    So, this element belongs to S 1 and this element belongs to S 2, so implies that this element belongs to S 1, so this element belongs to S 1 plus S 2. So, that proves that S 1 plus S 2 is a subspace.  So, we want to show that S 1 plus S 2 is closed, that means we have to show that f belongs to S 1 plus S 2, so that is what we have to show.  So, f n plus g n is a Cauchy sequence is a Cauchy sequence.  So, this norm goes to 0 as n and m go to infinity, but now note that f n minus f m belongs to S 1 and g n minus g m belong to S 2.  So, that is by Pythagoras theorem and this goes to 0.  So, meaning what this says that f n itself is Cauchy and g n itself is Cauchy. So, that implies that f n is a Cauchy sequence is Cauchy implying that f n must converge to some h. Similarly, g n must converge to some g. All in L 2. Similarly, that implies that f n plus g n converges to h plus g. We know that this converges to f, so that implies that f is equal to h plus g. Now, note that f n is a sequence in S 1 and S 1 is closed. So, this h belongs to S 1 and g belongs to S 2. So, this belongs to S 1 plus S 2. So, this completes the proof that if S 1 and S 2 are closed subspaces and S 1 is perpendicular to S 2, then S 1 plus S 2 also is a closed subspace. Finally, we claim that if S 1 is a closed subspace, then we know that S 1 intersection, S 1 perpendicular is 0. So, in that case L 2 is equal to S 1 plus S 1 perpendicular and the reason for that is because this intersection is 0. So, there cannot be S 1 plus S 1 perpendicular is a closed subspace. So, there has to be an element. If it is not whole, then there must be an element outside which is not true. So, that means for every closed subspace S 1 of L 2, L 2 can be expressed as S 1 plus S 1 perpendicular. That means every element of L 2 can be represented as an element in S 1 plus an element in S 1 perpendicular and this decomposition will be unique because S 1 intersection S 2 is S 1 perpendicular, S 1 is a subspace. So, there is nothing common between them. So, this is also called sometimes the projection theorem. That means for every closed subspace S 1 of L 2, L 2 can be represented as S 1. One writes as a direct sum of S 1 perpendicular, namely these two are equal and the intersection of these two subspaces is equal to 0. Let us come to analyzing maps on the space L 2 which is a vector space. As on any vector space, one can analyze linear maps on the vector space taking values in the underlying field. Here, our vector space is L 2. Actually, it is a Hilbert space. So, look at a map which is a linear map T from L 2 to C. We say it is a boundary linear functional if it has the following properties. First of all, it should be linear as a. So, T is a linear map as a vector space L 2 to C. Secondly, we want that it is bounded in the sense that if there is a constant m such that norm of T f is less than or equal to m times the norm of f 2. So, this is called the boundedness of the linear map T. So, we say T is a bounded linear functional if T is linear on L 2 and absolute value of T f is less than or equal to a constant m times f, where m is a constant fixed and this happens for every f in L 2. It is quite easy to this condition boundedness is actually it implies that T is also continuous because if f n converges to f, then T f n absolute value is less than or equal to norm of m times the norm of f n minus f and that will go to 0. So, it is easy to verify that T which is if it is bounded, if a linear map is bounded if and only if it is continuous and because you are on a vector space continuity at 0 is enough. So, one can verify easily that every bounded linear map is continuous at 0 is equivalent to it. One way of defining a linear maps continuous linear maps is the following. Well, for fix any g in L 2 and look at the map T lower g defined on L 2 to be T g at f is equal to f, g for every f. So, that means that the value of T g at f is defined as f g inner product of f with g for every f and it is easy to see that this is a linear map because the inner product is linear in the first variable. So, that will give that it is a linear map and it is bounded because of the Cauchy Schwarz inequality. So, this is linear and by the Cauchy Schwarz inequality it is a bounded linear map. So, one way of constructing bounded linear functionals on L 2 is by taking the inner product of any element f with a fix element g and this is important theorem called these representation theorem which says that this is the only way of constructing bounded linear functionals on L 2. So, it says that T if T is any bounded linear functional then there is a unique g 0 belonging to L 2 such that T f is equal to f g naught. That means every linear functional on f or linear functional T on L 2 arises via inner product of f with a fix element g 0 and this g 0 is also unique. Let us just outline the proof of this. So, first of all let us observe that there are two cases look at suppose that there does not exist any g 0 with the required claim then what will happen then g 0 must belong to what is called the kernel of T of g. So, that means what and kernel of g is all elements say that which are mapped to 0 and this is a closed subspace of kernel of bounded linear functional is a closed subspace. So, if there is no g then this will be so that means that our required claim will hold with T equal to 0. So, that is essentially saying the kernel of T g is a closed subspace of g and if one possibility is kernel of T g is equal to the whole space then it is equal to 0 because if kernel of T g is equal to the whole space then T g will be identically 0. So, any g 0 will satisfy 0 equal to 0 will satisfy. So, let us assume that the kernel of T g 0 is a proper closed subspace of it then by the best approximation theorem g 1 there is a g 1 in kernel of T perpendicular that is the consequence of the best approximation theorem and thus g 1 will not be equal to 0. So, in that case one verifies that if you take g 0 to be equal to T g 1 divided by g 1 comma g 1 into g 1 if we this selection of g 0 is a required unique function such that T of f is equal to f g 0 for every f in L 2. So, essentially one applies the best approximation theorem to get an element g 1 in kernel of T perpendicular and why one is looking at kernel in g in kernel of T perpendicular is because if the required condition is to hold then that function g 0 has to belong to kernel of T perpendicular because if it is g comma f and that means for f in kernel that must be 0. So, that means the required g 0 has to be from here. So, let us pick up any element and then modify it and show that that is the required. So, this is what is called the Ries representation theorem. So, this is the Ries representation theorem. So, today what we have looked at is the space L 2 is a perfect generalization of the space of the R n or the space C n that means there is a notion of a inner product defined on it which gives which is related to the norm and which gives the notion of perpendicularity. So, we prove we stated one important theorem namely if s is any close subspace of L 2 and you take an element f in L 2 then there is a best approximator then there is an element g 0 in the space close subspace which best approximates with a minimum distance from f and as a consequence of this one consequence is the projection theorem namely every close if s is any close subspace of L 2 then L 2 is a direct sum of s plus s perpendicular and the second consequence is characterizing all bounded linear functionals on the Hilbert space L 2 namely linear the only way bounded linear functionals can be constructed on L 2 is by the inner product that means t of f if t is a bounded linear functional then t of f must be equal to the inner product of f with an element g 0 for some element g 0 with the inner product. So, that is characterization of bounded linear functionals. Thank you.


The Project Organizer is an integral part of your dashboard and displays your project components and registrations. From here, you can navigate to any of these parts of your project via this light blue button. It also makes it easy for you to arrange them into folders that make sense for your application. Your projects and registrations will initially be shown in smart folders called All My Projects and All My Registrations. Of course, you can create other folders to meet your own organizational needs. For example, you may have so many projects that the list is overwhelming and confusing to work with. In this case, you may want to create a folder containing only the projects or components you work with most often. Or you may want to be able to access your analysis scripts from all of your projects in one place. You can make a folder called Analysis Scripts and add only those specific components. After you create a folder, you can move them around and rename them by clicking the folder and selecting the Edit button that appears in the top of the Project Organizer. Dropping projects or components into folders doesn't change the structure of the original project. It just makes it more accessible to you and your contributors. Likewise, deleting these folders does not delete the project and components they contain. If you want to delete a folder, simply choose Delete Folder. Don't worry, doing this won't delete your projects or components.


And thank you for returning to Adventures in Commercialization. This week, I have the pleasure to be with these lovely ladies. I have introduced them as members of my company, onboarded them to just meet and greet. But when I really dive down deep into what they're doing, it has inspired me to want to bring them on this show because they've done an amazing job to promote their business. They've pivoted to the maximum, and we're going to learn all about it today. So welcome, Mickey, and welcome, Sarah, to our show. Thank you. So ladies, tell me a little bit about yourselves. Tell me how we got here, how we met. Let's dive deep. All right, so we'll start at the beginning. Sarah and I were trying to decide if we were going to disclose how long ago this was. But almost 20 years ago, we met in our very first day of college freshman year, and we've been friends really ever since. And we became business partners about a year, officially about a year ago. A year ago, and you guys done so much in such a unique way. Wow. Then a roller coaster, yes. So I know we're on think tech Hawaii. And I learned a little bit about what Sarah has been up to. And Sarah is really excited to be here today because she actually began as a cheese connoisseur in Hawaii, which is great because I grew up in the Netherlands where cheese is like a bang, like the Gouda. So yeah, tell us. I'll say, we call ourselves Cheese Mongers, the official title. OK. But yeah, that's a nice ring to it, too. But you know, early 20s, living in Maui and working at a wine and cheese shop, it's interesting to think about what I didn't realize skills I would learn there in the business and from our female owner that now I use in every day 50 post life, much, much later. So cheese connoisseur or what was the word you use? Oh, cheese monger, cheese monger. All right, I love that. So what were you doing in Hawaii? So you said Maui and Maui, yeah, and Kihei. So it's a retail shop, woman owned, small business. So a lot of self promotion. It was early days, Facebook. So trying to use that as a marketing tool. And we had newsletters. And so I did a lot of dealing with vendors, with purchasing of the product. We had shipments going to different islands, people shipping product back to the mainland and just kind of learning business. I had never worked in a small business. And so it was just a whole different field than working for a corporation. I'm a lot more hustle. Hawaii is a perfect environment to understand that procurement understanding.  You have to understand the locality and where you're at, what their local resources are, but also procure things, get them there, shipping, facilitation of that like liaison type style of management. But that can just go into so many different sectors. And that's what I would really like to tell people today is that all of the things that you've done in the past, they can definitely push you towards your future. Working at McDonald's or service industry can help you become your own business owner. And little did you know. Or cheese shops in Hawaii. And now we're doing a talk show for Hawaii success. But you ladies are the art of pivoting.  And meeting people in your past. I have girlfriends that I met 15 years ago at Girl Scout Camp that we are still friends to today. And we do art projects together over Zoom. So it's really great to understand that pick up the pieces of your past a little bit. Like take those breadcrumbs that you've laid down and keep them for your future. Keep them in your back pocket because they could totally create a successful business. So tell me a little bit, Mickey, about what you're doing now. Yeah, so I'll give you the short version of how we went from cheese monger. And I spent about 12 years working in London and moved back here and produced a storybook. And as I produced this book, started to sell it. Starting my first solo business there. Sarah got involved and helped me with packaging. And then we started to add some spices to the books and people liked those. And then we found ourselves with a gift box around the book. And then in the meantime, because of COVID, Sarah ended up losing her job at Vulcan where she was doing procurement. So again, to your point, Zoe, about going from a cheese shop, like how do you go from procuring cheese to procuring airplane parts for a billionaire. Sarah transferred those skills and then was helping me with storybooks. And then suddenly she realized, this is a chance to kind of do what I love and just be more artistic and with food. And so then we ended up doing gift boxes. And then Sarah just was with me all these gift boxes and realized, actually, this is the business. And then what was Pixie Spice became Pixie Post. We launched a new website, as we said, about a year ago. And we took all of that early customer success and then built it into what has now become a repeatable business based on just what people are gonna buy. So it's been a really interesting experience. So you're both in Washington now? Yes. Okay. How do you market yourselves? Like we talked about Facebook before a little bit. The biggest thing about the show and the reoccurring instance is like people who lost their jobs to COVID. I'm one of them, Sarah, you're one of them. We are people who had to pivot hard and pivot fast. But it's a blessing in disguise a little bit here. So take your passion project and turn it into your livelihood.  Yeah, and that's like one of the decisions that COVID kind of took away for a lot of people. It's like, here we are, I've got a new thing. Do I go and try to find a new career or do I use this little bit of time in this new environment to see where my passion and dreams could take me? And so for marketing, really Zoe, we've tried different paid marketing tests and we found that the most successful marketing for us has been word of mouth referrals. So we're really lucky to have a strong network here in Seattle, which has then brought us business all across the country because people will send gifts that will mention like, oh, I bought this wonderful specialty gift box for my grandma from Mother's Day, but also I'm in a team of 15. And I want to tell them that they did a great job this quarter. And so I want to do a specialty gift for them. And that's just kind of how it's grown and grown. And, you know, we met a few months ago and I learned about Wilchana, where assistants help busy parents deal with all the things. And that was another example where we got to pivot is that we had a conversation and you said, you know, actually, we might not be interested in buying all the products, but we might be interested in the wrapping and the packaging and bows and wax seals and things that make the recipients feel like this is a really special thing that's coming in the mail to them. So now we have a wrapping service. On the totally other side, we've started to do housewarming presents for Airbnb hosts. So instead of doing all the packaging, we do the sourcing and then we can help them buy in full and have that all set up for all their guests for the summer. So we're going from gift box to only gifts or only box. Oh, dang girls, you guys were just doing it all. I love it. I love this art of pivot. It's so important. Something that you could take from a passion project to a career, but also understand that it's not just a linear view. It does have a lot of avenues that are willing to make money here in the future. So pivoting is so important and being open to the ideas of something new is very important.  Do you wanna be too innovative? And they're like, no, well, you have to hit the milestones but you also wanna be something new and like exciting for everybody. But you ladies have just been the art of pivot. Literally like, oh, this girl's talking to me about wrapping presents. Well, we do that. So why wouldn't we get paid for that? Let's try it and it's great. So I absolutely love it. So what kind of milestones are you looking at in the future? Where are you at now? And what things are you trying to hit this year? Oh, that's a good question. Sarah jump in here to you. I'd love to be able to hire a person to help us with all of the process stuff. It's like Sarah and I will spend hours coming up with ideas and keep up great packaging, doing staging, coming up with ideas for our social posts. But man, the hard thing about a small business is the taxes, the accounting, like making sure that we are doing the right level of analysis about the profitability of our different product lines. That kind of stuff is hard. Nobody wants to be an accountant. Okay. Just leave that to another job. I also have a problem. But so you ladies are working together but out of your homes, correct? Yes. Yeah. So you have like an attic full of wrapping paper right now. Okay. Yeah, my whole second floor has been converted to PixiePose. So if we can look at the PixiePose website a little bit in the background here, we will see what you guys have done, which is absolutely phenomenal. You two are just a success story here. All of that wax seal, Zoe. Can you scroll back? I know. So a little bit of background story is Johanna. We have not yet had Johanna on the show, which I'm looking forward to in the future. However, Mickey was one of the first members that we had at Johanna, which is a startup app to create more wellness and opportunity into the modern family home. And to give them more time to focus on their passion project, and to give them a little bit more time back in their day. So Mickey was doing this PixiePose. We asked, we have one member on the show or on our membership that does gifting. He literally sends 60 gifts a year. I'm pretty sure that's probably the only business you've gotten from us. But he literally sends like 60 gifts a year and he's like, I order them from Macy's Nordstrom, but I don't want them to look like they came from Macy's or Nordstrom. I want them to look like they came from me. And there's not really a concierge service that does that. There's not really somebody who takes it from A to B and makes it look like it came from me. So we switched that up and Mickey and Sarah have pivoted their company to create a customized seal, a handwritten letter. Absolutely, like, there's no touch points that you need to participate to make sure that everybody in your anniversaries, birthday gifts, all of these distant relatives that you wished you paid more attention to, these ladies are doing it.  I just literally had to write down my grandpa's birthday because I didn't even know when that was born. Now, if you give it to your assistant, PixiePose will take care of you. Okay, they will give it. And you guys are also sourcing very locally. That's your main idea. I mean, of course, if you have a co-worker that's moving to Florida, you ladies will locally source in Florida to make sure it feels good. It feels right. It feels comfortable. It feels to the theme. So tell me a little bit more about your sourcing of your product. Is there what you want to share? You know, it's one of the fun parts of I think of the job is just vendor relations. It's something I did a lot in aviation and it's something I still really enjoy. And especially working with the small businesses, we try to source a lot of Pacific Northwest products as we're in Washington and Seattle, so in Washington, Oregon, Idaho. But as you said, Zoe, we do source out throughout the nation the recipient is out of date. We love doing customization for our clients. And so for sourcing, you know, our values are small business, women-owned, minority-owned, handmade product, which there's a lot of, especially in the last couple of years because of COVID, a lot of businesses have popped up. And a lot of people are now making products. And it's been really fun to try and make those connections. So anytime I'm out somewhere, I'm always keeping an eye out for new product, snapping photos and sending emails, seeing if people are willing to do, you know, wholesale orders for us. Sometimes the business is way too small and they say they just can't make that much product. And then a lot of times they can't make enough for the orders we have. And so that's been a limitation to our value system, but it's still something that we want to stick with because we believe in it strongly. Right? All those Washingtonians with those blackberries in the backyard, so if you're up in your businesses or I don't know, Hawaii, you got blueberries out there, raspberries, we'll take them, okay? Absolutely. I would love to do a Hawaii box, Sarah. I know, I've been plotting one in my mind. I haven't executed it yet. I know, we're in the right place for the right time. Watching anybody out there, email me Sarah at pixiepose.com. PixiePose, you've been a product. We're in little jars of something, we'll take it. We'll put it in our life. Absolutely. And Zoe, I think one of the things that's really important about what Sarah said is that as a small business, we understand what these small businesses are struggling with in terms of distribution and reach. So we'd love to be able to help a great product get more users and buyers from across the country. So that's one of the things that really motivates us and makes it worse, sometimes the extra hassle. Or we have to sometimes ask our customers for some patience because the really great product that we want is literally being hand-poured in someone's kitchen and they're hand-pouring 300 candles for us, but they might not be able to do that in two days. No. And overall, our customers really responded to that well. And the gamble that Sarah and I are taking is that that extra work of sticking to the small business support theme is also allowing our customers to stick with us too because they know that we're consistent and we stand for something and we help them shop their values too. That's great. We all have to start somewhere and we all have to promote each other. So if you guys had a couple of, let's say, hurdles that you faced or, let's say, advice for other entrepreneurs, let's say women in specific because you ladies are doing it and doing it well.  Sarah, do you want to start? Sure. I guess for me, just starting this business, it has been really, really important and really impactful for us to rely and for me to rely on connections that I already have, rely on my network, on my community. As a woman, I was in the sorority that Tom Mickey and I met and so relying on those female connections, that sisterhood and then using skills that I've learned in the past that I didn't know could translate to the future. So don't underestimate your abilities and don't be afraid to push yourself forward. Correct, I totally feel that. I literally just went to lunch in Seattle with my coworkers for the first time we went IRL in real life. Oh wow, how nice is that? And this is the first time I've worked with all women. It's crazy. Like I used to work in the startup seasoning or feeling and that's all like middle-aged white men, right? Let's just say it, but I actually was working with all women and I got to meet all women in Leschi with a little bit of sunshine, wild, astronomical thing to think about. But when we met each other, some of the people have come from corporate Taco Bell and know about nutrition. Some people came from the medical field. Some people came from, I am a corporate event management, but anywhere you came from aviation procurement, which is really crazy because my parents are aviation mechanics. I don't know. They're Spanish and Southwest and American. People who came like from that. So if you think about the skills that you have, like take it and use it. Don't just think it was a job that you could use, but just like think about how those skills are utilized in the future. Use them because they are real. They're hard skill. They are hard skills and you could create an entire business from it. So you ladies are doing great work. Your pivoting skills are phenomenal, like phenomenal. And you're doing great things and they met each other in the sisterhood. So your network around, that was another episode that we had was talking about network, networking. It's so important. Take those people who have hard skills from your past, from your future and just utilize them and just think about how they can pivot. Just pivot them. Just take them and move on. Absolutely. And it can sometimes feel like you don't wanna sell your friends, right? You don't wanna just feel like the only thing you're doing is be like, hey, I've noticed you haven't ordered anything. So Sarah and I have been really careful about balance and trying not to oversell. But when we've talked to our friends about it, they've been like, no, you need to remind us because I literally forgot. So I think that's as women, particularly, it's easy to just be a little bit less. I mean, you could use the word assertive or aggressive because we think it's going to be perceived in a certain way, but we check in and get feedback and find out if you've gone too far one way or the other. And what we more consistently hear is we wanna hear more from you. Tell us more instead of leave us alone, which I think is a good sign. Well, you guys are doing so well. And honestly, actually one of the criteria I have for dating a man is he has to be a good gift giver. So. Good one. That's a good one. You have to. If you have a friend that's a good gift giver, just make a business about it, you know? You ladies have done so well and I'm so excited to see where you go in your future. You're doing things, you're promoting local businesses, which is something that just needs to happen. But where was the place where you said this is gonna be my full-time jam? Like where did you just say like drop the day to day and become my full-time jam? When did that happen? I think for me, I don't know that I can fully pinpoint a timeframe, but I guess maybe if we put it last the holiday season because our business is very seasonal and the holiday season rush is so exhilarating. And I was just having so much fun and it's just really great to be giving gifts for a living. Somebody else is paying for them. I get to put them together and then mail them out and I'm bringing joy to all these people across the United States.  I had just a great time and I thought this has to continue.  And she's like, she's such a great gift-giver. Let's make this money. Yeah, and my role is more but we have to charge people for it. So I mean, that's another piece of advice is that it's easy to get swept up in a good product but you've got to keep an eye on margin because if you don't, you don't have a business. So your financial advisor is Mickey and you're just the type to be like, okay, we get it. Now, because I would want to try and give everything away for free just because I think it's so fun to put files on people's faces, but yeah, Mickey's there to say, we're a business. I love it, because you have to have a practical person and an emotional person, like motion-giving and the person who's like, okay, there's money behind this, let's go.  So I love that.  We've built up a pretty good repeat client base from a few sort of agencies and businesses that want to celebrate customer success or that want to do onboarding. And that for us is a really good balance to the seasonality of holidays on our consumer gifting side. So like Sarah said, the holidays are great because we sell a lot and it's very busy but then January comes and it's like, where did everybody go? So for us this year, getting more even revenue across the year and trying to add in these other bumps. For example, creating our own seasonality a little bit and promoting Airbnb host boxes or host kits at the beginning of the summer season and like just trying to get some more of that to level out, I think is a big goal for us and it's important as we move to the next stage. And the Airbnb thing, we didn't even discuss that in our little pre-talk here but phenomenal pivot for you too. Oh yeah, and it's been a really cool one actually because it came from an idea that one of our clients had for an employee gift box that was all about movies. And then we were talking to some other of our friends and family who have Airbnb's and realized that this movie box, which we started to sell and is sold really well on our website, we could turn it into a popcorn corner for Airbnb. So Sarah got like set up this beautiful little popcorn for, yeah, that's it. And so there's like a little popper, like this really awesome salt from San Juan salts and like a little sign in a tray. And then for the Airbnb hosts, the cleaners just have to replenish the popcorn and they've got that there for them. And then we also have some like customized toffee that has a picture of the house that they're staying on it. So it's really fun. And I think there's a lot of mileage in that too. I think you're right. Wine vineyards and you guys with your face on the like wine bottles now. And like all the time I'm thinking of my parents, actually they're the biggest homebodies you could find. They are literally have like, they work for the airline. So when they're at home, they just wanna watch movies. So they have a home theater, like they could just have or just you two are the art of pivot. Like literally you'll send it in whatever direction that wants to be specialized. You're the ultimate gifters, but the ultimate pivoters. So really fantastic. I think you ladies are doing a phenomenal job. And I'd love to bring you back on in the near future. So we can just see where you're at then. But if you had one last piece of advice for entrepreneurs in general, what would you have? I would say don't be so hard on yourself.  So just believe in yourself and give yourself a bit of a break. And I think mine would be kind of in the regards of pivoting is don't limit yourself. Don't have boundaries, be open to new directions and just see where that goes. And you just never know where it might take you. Well, I'll take Sarah's and put it back on Mickey's cause here we have a beautiful mother. Both of these two are beautiful mothers, young ladies in success that are here pivoting their life, met each other as friends, turned into their reality, passion to project, okay? And then at the same time, Mickey was one of my members coming in and it's okay to ask for a little bit of help sometimes but came in saying that she needed a little bit of help with the home front of I'm working on my business. I'm a working mom pivoting to just try to make my sisterhood project work and I'm here to help her in the background. And at that same time, I was like, hey, can we make you more money by just taking like your big idea into like a little bit of sectionalized idea and turn into just rapping. And again, it's very, let's say seasonal because I know you guys aren't making that much money on the day to day, but there is that one guy who just wants to buy Macy's products for his members with a personalized little standpoint. And that's probably where you're getting all your money from but there is just a pivot that just says we do this but we will do this. Yeah. Take this to another level. Well, if one person is gonna buy it, there's a market there. So let's promote it and hopefully get a thousand people to buy it. Well, today I just opened LA chapters and you guys are on our member lists. So we're going, we're going to the distance. We're gonna be the new Ubers of assistance over here YouTube gift wrapping gurus. So I absolutely love the fact that we got to connect more. I hope to watch your progress going here in the future. These two are the art of pivot and the art of exploration and just find somebody who's in your past and take your past to your passion and take your passion to your passion project and make it a career. Love it Zoe. Thank you so much. You make us feel good about ourselves. We really appreciate your support. Always from cheese to aviation, from Hawaii to Seattle, let's do it. So thank you so much for coming on the show. We look forward to seeing what you ladies are gonna do in the future. Thank you Zoe. Can I just say www.pixie-post.com? Yes, go send them your birthdays, anniversaries, any seasonal without even seasons. If you send them the dates, they're gonna send them a gift. So we're gonna make this happen. If you wanna learn more about how to make money, join us every other week for the adventures in commercialization. And we're gonna learn more about entrepreneurship. But thank you so much for your advice. Thank you so much for watching Think Tech Hawaii. If you like what we do, please like us and click the subscribe button on YouTube and the follow button on Vimeo. You can also follow us on Facebook, Instagram, Twitter and LinkedIn and donate to us at thinktechhawaii.com. Mahalo.


Let's seal the breakfast and plus Xavier Africa and thanks for staying with us this beautiful Friday morning. Gideh Johnson is on standby. He will be sharing his thoughts on the papers. It feels like there's a lot of congratulation and jibberlation like some of the papers would say. Gideh, it's good to have you join us. Good morning medicine. All right. Good morning to you. Good morning to you Mr Johnson. Maybe we should say congratulations because you're your lecture as well. People we have going around as the decision of us. Congratulations in order. Well it is in order at least that our children going back to school do not be roaming the street and it's very clear that we cannot have 100% of the students that were in school then going back to school who have lost some to one thing or the other. Some might have even lost interest in going to school again so well but it's better at least than believed so it's better for us to have the school reason than for it to perpetually close. So congratulations to you and I, particularly congratulations to the student whose life has been concreted by the insensitivity of government and to listening to the plight of ASU of the student that represented by ASU because overall what ASU is asking for invariably we affect and influence the welfare and the training and the discipline of students in various tertiary institutions. All right then let's start off with a punch newspaper. The punch says Namdi Kano jubilation in southeast federal government may appeal judgment and I don't know if the jubilation is just in a certain region or a lot of Nigerians are happy but that's how the punch captions it. Codd says IPOP leaders abduction on lawful slams executive for recklessness and it took a lot of time for you know the realization. Ohanese Sore south east elder statesmen Kano's followers hail judgment. We will present judgment copy to DSS for Kano to go home says or ozakome that's it. On the punch the board caption and the riders underneath it. Away from that World Bank IMF to access Niger's debt sustainability PDP was aggrieved party members BOT delays reports and strike parents students hopeful as ASU leaders meet but I'm sure that you know when the punch actually was set and done for this paper to be put out ASU was actually having that meeting. It was like you can say when wild men slept ASU and the federal government were having a meeting. So who gets federal government's approval for five for for 2.5 billion but Agri C port. So who gets federal government's approval for 2.5 billion dollars but Agri C port and just before we move away 126 stranded Nigerians return from Libya Legos probes nurses for filming ex big brother Niger stars last moment and it was part of our conversation on a top trending this morning. Osinachi I thought her abusive husband would change that's what the neighbor is saying I thought her abusive husband would change but what what does all of this mean now that she's gone that's the much that we can take on the punch newspaper. All right let's go quickly to the nation the big one taking the most of the space on the front page on what technical core others can't tackle bandit recess arrow five he's been speaking a lot recently he says it's simplistic way of fighting bandits or Tom people have final say on kitting outfits with AK-47 let's see how he goes on that more from the nation ASU in all night meeting over strike well I'm sure we know the outcome of that meeting and hint was given already during the last meeting between the ASU leadership and the House of Representatives speaker. The ASU president said he's seen a light at the end of the tunnel. I'm keep asking what kind of light did he see anyway it didn't show him South South Commission Zokoa Diri to pacify weekend will it work Sao Lu signs MOU with Del York film city on film city rather a court acquits ex governor to Raki of 8.3 billion Nara fraud congratulations to him NPC backs local tech firms to par 2023 census 2023 it's reward time for Tinibu say quara governor senator appeal court ends Namdi Kano's trial orders IPOB's release or uneasy it's best news of the year he has not been acquitted says Malami of course the attorney general and minutes of justice saying that it was he was only discharged but not acquitted and saying that the federal government will explore every legal avenue to ensure that other pre rendition matters it calls it pre rendition matters are taking care of so it's probably it's not the end of this case LP's presidential campaign begins October 18 in Nasalwa and Baku remains APC members are stories on the front page of the nation well let's move away from the nation we have the leadership out as out of school children's coach worsens Nigeria ricks losing out on literate and skilled workforce so you have a representation more like a graph right there if I'm not mistaken 6 to 11 we're talking about the ages now the age range for out of school children in Nigeria so between 6 and 11 Nigeria Indian Pakistan countries with the highest figure out of school children again you have 35% as estimated you know number of Nigerian children who attended primary schools do not attend secondary school 10% almost 10% of Nigerians 216 million population are headed for lifetime literacy and 60% almost 60% of girls in this court not in state out of school children especially now that you know we just recently celebrated the girl child 85% of jobs that will exist in 2030 have not yet been founded wow it's a lot I'm thinking that this is a lot of information that we need to you know pay attention to whether or not you can say this is true but we should be working towards this direction we can increase this political age to 50 000 insist on sadden PDP chairman just how many months before the elections critics cannot forge governor Yahya's giant strides in Gumbay that's what an aid is quoted to say a pure court acquits and discharges Kano of you know terrorism offence 2023 censors nmpc or npc inspects factory for local content that's the much we can take this morning and nigeria has not asked for debt restructuring according to the world bank all right moving on from the leadership on friday we will take the last paper we have the guardian with some interesting headlines to touch on a few of them ipo b must sob jubilee this court cautious charges against Kano's lead story on the front page of the gallery and the following writers Kano discharged daughter quitted fg to explore other options court court fg subduction rendition of Kano um breached our local international laws Kano's freedom a passive step towards biafra restoration must sob uh that's the movement for the actualization of the sovereign sovereign state of biafra uh ohanez and digbo fg should not appeal Kano's victory it's time for dialogue as what they're saying i think uh ohanez they didn't waste time to send out its press statement last night fresh concerns on state of basic education is out of school figure reaches 20 millions very important story and they call it the big story they have details on pages four and five of today's edition of the guardian with a number of pictures that are really uh tell a story tell a story doubts as nigeria negotiates debt relief with world bank imf uh chimamanda did not accept international honor confirms a team confirms she was conspicuously absent and i think basically we we asked the question or two about this yesterday we went to our guest somo lu del york sign mo u to build ligas media media a film city confucian at aero contractors is amcon backpedals and ng eagles plan amcon backpedals and ng eagles plan we case a donation stairs controversy in cross river state look at that as we go on it's of interest to me um and these are some stories on the front page of the front of the guardian um over to you jane johnson what are your thoughts on the the development with namdi canu as far as the federal government is saying the case is not yet over and uh abu waka malami at nigeria's attorney general um specifically said that they issues that uh up for judicial judicial determination issues that predate his rendition he called it pre rendition issues that uh uh the courts need to determine so it seems this is not not yet a hole for um carol well we should issue if it's if it's in other time it will have resigned and perished and um the courts deal with process and procedure and if the later process and procedure are not followed um you lose the case on technical grounds and if you see um the judgment yesterday by the epic which was unanimous it's about um the federal government not following the damn procedure not having respect for human right not following international treaty that nigeria is part of parcel of you can see the court condemning executive um recklessness abuse of power and privileges and accusing government of engaging in in non-state actions of abduction and kidnapping if you look at that no matter what case you bring before the judiciary if you don't follow the normal process and procedure you lose the case you you you you you you you lose the case on technical on technical ground i think um like others have advised i think that which you look for a non-judicial settlement to this matter that's that's the way which both parties can be able to to to save to save their face if you continue with this with this trend if it goes to the supreme court the supreme court is going to throw it out as because you have an unanimous decision at least you see the supreme court is agreeing is agreeing with the appeal court in in cases where you have unanimous decision so as far as as far as i am concerned like pointed out he has been discharged he has not been admitted because there are still legal options that the federal government will still explore to prosecute to prosecute their their case but cases are won on the basis of evidence cases are won on the basis of process and procedure and if you look at what the courts in terms of criminal police judgment don't just sledge armor that the court was using to to to to hammer the federal the federal government interesting interesting but but i fall up to that yes i fall up to that um if kindle manages to make his way out of uh detention um it's been referendum now as far as he's concerned um should he continue saying referendum now so should he and his group uh change it to election now looking at what is around the corner and i think i think what i think any idea and that we actually want to progress of we call for us to go back to to pre pre-1966 era to go back to regional government whereby the bulk of the power are with regional government and the central it's whatever the region contribute to the federal that the federal use not the situation whereby you take the resources from the region and then you take the resources to the future and further will not be distributing the resources in piece mean to do until we do that in the simple word for that is what people have called restructuring until we do that no matter who you are late as the president of nigeria the president cannot solve the problem of nigeria because we have a foundational problem with the structure of our governance with the structure of public administration in nigeria the structure is there now let me take it to use another story in the newspaper you see the lego state government to go to economy you see lego state government is seeking the approval of federal government to construct a seaport in badagry federal government is in a budget what right does federal government have over per badagry and maxing is badagry local government under federal government over the government under lego state government and i'm telling you until we practice true federalism and we allow each of the region to go back to pre-1966 structure i can assure you whoever becomes the president they will insult him like the insultant duary like the insultant tunata like the insultant of asadov the only person between our time to insult was was yadda even yadda his wife was insulted was divided over time so the as long as you see for you zikziklase for you to do the same thing the same way the same manner and expect to get a different reason is the beginning of insanity now we are close to 24 years of democracy are we making progress just so jina just a question before we move away from this particular issue let's look at all the parts of it because some people say it's important that we pay attention to antecedents now in 2019 we remember that namdi khanu surfaced miraculously that's what some people have described it and when he he you know he arrived it wasn't too long before he also was asking for a boycott of the elections that 2019 election asking for referendum and you know that at the point where he was asking for this election was a struggle for the opposition now the question a lot of people are asking is why now the timing if this is what you know the cut of appeal have decided to do why now same five months to the elections are we seeing a reputation just like kofi is asked is there going to be a call for referendum and we understand what that would mean you know the fear of not having people come out to cast their votes the fact that you have a certain region wanting to opt out so is this politically motivated motivated or it's just you know the cut of appeals decision do we have the chief will for the national assembly the senate is on the south east do we have senators on the south east do we have as a red member from the south east we have been referred to executive cabinet in the south east what is the political trust and direction of the south east as represented by the elected appointed representative these are the easiest if you look at there are ways and means to you could go about your annotation they could they could come all the south east senators and as a rep who come together and propose it be come up with a resolution or what have you have they learned that you see calling someone can see enemy and call for referendum if you don't have the the except you are except you are putting except you want to use the putting approach that is using that is using in russia and that will not work that that will not work in this in this in this in this contest as far as i am concerned there are still means and instruments through which we could act for restructuring of night that when you talk about the referendum or want to partner with or whether we should move from federalism to confederation to become a confederation okay you know what each of the region would be independent but for security international trade and what have you come together to form a confederation like like scissor land no problem both you have divergent and discordant troops coming from the east none of the candidates representing one interest group the elected representative are representing one interest group peter b two is representing an interest group would peter be support referendum now will he support the cancellation of the election i mean it's just very interesting to i mean look at it's something that has happened before so there's a lot of worry and concern whether there will be a reputation and what would they make you know for the electoral process or the electioneering uh in 2023 especially for a certain region and for those who they might just be supporting but um do you know we need to message journalism is each period in memory the future is clearly the present i cannot show you whose 2023 election the agitation for safety determination will be high i cannot show you that will be high all you just need to do is just to look at the the the the political atmosphere the social atmosphere and what everybody every region as they have invested their hope on this electoral process this electoral process that has not changed the fortunes of Nigeria will not change the fortunes of Nigeria 2023 the structure is 40 if the foundation is not solid there's nothing you can build on it as long as you continue to build on the 40 foundation the building will collapse and that's why we have seen the collapse of public administration nightmare i'll take you to another story you will see that the failure i caught in Tutsi in jigawa destroyed and acquitted samino turaki the traveling governor no a case that the start is 2007 the court ruled yes the anti-target man acquitted of 8.3 billion okay before we run out of time jenny johnson the economy is um is of importance to nigerians and uh we're looking forward to seeing what will happen uh with the 2023 budget where the money will come from um the guardian has his story doubts as nigeria negotiates uh debt relief with the world bank and imf um i've been monitoring events in in the ghanian economy with discussions um and the ghanian government going for this imf and world bank uh bailout and all that and it's been a hot topic like people are talking about it and both for and against um it seems like the federal government of nigeria is going to saunter into some sort of deal with these institutions without nigerians even knowing anything what are your thoughts on what's going on because um the finance ministers is closed um that uh the federal government exploring the debt restructuring options as well as uh securitizing the 22 trillion central bank of nigeria overdraft you know and they are at the world bank imf and general meetings as we speak well kofi um at the beginning of this democratic experience we were able to secure debt forgiveness secure debt forgiveness we even have resources in our foreign reserve and when this administration came into power we have been caught debt that we have nothing to shoot for it and if you ask them question they will show you that they have done infrastructure and you actually add infrastructure that they borrowed this money to to to build it's left for nigeria whether for them to reinforce video or to reward video because i i can assure you i i i can't see anything where with what we have done with the debt you have been caught when in actual sense when a person was living when you are where was living when you not can't was living we had money in a foreign reserve the foreign reserve had been depleted that we have now become indebted to imf and wopaki game probably an economist will explain it will explain it but in in my simple apc micro economics i do i can't i can't i can't understand how we found ourselves we did this weight of eight years with this remote cost debt and with nothing to show for it maybe it's because maybe because people have people have been asking questions i mean we we've had several budgets and we look at the amounts of money is being being put on on in the document you look at the one even for this current budget look at how much is being you know set aside for presidential travels for the presidential fleet how much has been set aside you know to to maintain asu rock for instance and then the national assembly is just going to pass it and people are just going to keep quiet and we'll we'll sometime to deal with the world bank and imf and then maybe we have another sap to add to what we're having as the economic world we got two years ago the budget of two years ago where the national assembly voted huge amount of money i don't know whether it's six billion or or what have you to renovate the national assembly to renovate the house that did not he built it that was commissioned i think the second one was commissioned when i mean david mark was was was the senate was the senate was the building is not up to 20 years they renovated it for nature only that is the reason why let each region keep its resources and let them contribute to federal government there will be less wastages well so the question is how can it be although we know the answer already who's willing to take the action for us to restructure the economy as it is jidya johnson this is a quick one because i mean that the agenda the media should be set concerning 2023 these are the agenda we should set these are the you see if we continue no matter how long you travel on the wrong direction in the wrong direction rather you will not get your destination no matter how long moving sometimes moving forward is going backward if you are going the wrong direction and you are going forward you think you are going you are going backward because you don't get your destination so moving forward from you will be to turn and go backward that's and the the reality of it is that if you want to look at when Nigeria had meaningful growth when Nigeria had strength both internationally and locally when the economy was better developed when the easy corruption was dealt with decisively it was when we had regional government and until we restructure you take all the resources when our government would take 57 percent of of your resources one one one unit one power center and then 30 33 percent will be for 776 local power centers 36 power centers at the state it's not it's so so we need to move away now but i i understand what you mean when you say we need to change the focus of our conversation but you also understand that it wouldn't be on us to actually have a constitutional review to say okay we need to get back you know to original kind of government or maybe ensure that resources or there's decentralization and resource control becomes you know the way forward but let's quickly look at this i'd like to share your pulse on this one it's on the leadership it's like saying the leadership is saying nigeria ricks losing out on literate and skillful or skilled workforce that's what it is the future the leaders of tomorrow don't have a feature today and that's because nigeria is topping the chart with india pakistan of out-of-school children and then you have about 60 percent of girls who constitute especially in core not state states who are not in school the statistics i'm sure you heard it when i went through it but what are your thoughts on this one does this scare you well um we call us to restructure about equipping our children for the future it is not everyone that we have formal university education we need to invest in technical education we need to invest in education that will give people technical skills from the when we are giving out the statistics we talked about a lot of them after primary school that will map the end of their education adventure educational adventure what what system are we putting place to ensure that we get we get technical technical labor force from those category of people no if you if you are looking for artisans in nigeria now you know where the artisan comes from they come from republic company they come from to go if you are looking for the best nuclear the tylas water it is the question i asked i don't know it was on it was on your station we did the international teachers did and we were talking about education and i said look a lot of nigerians have traveled outside of nigeria what type of jobs do they do jobs that they will not literally do in nigeria today do you know that to travel to great prision and you are great clear or you are plumber or whatever you make more money you make more money than for you to have a bsc or ms degree so this critical sector this critical support sector that provides technical education we need to make investment in it and the emphasize this this case for certificate certificate at the tertiary level you see there are different levels of education you have primary you have secondary then you have technical and then you have university education so we must look at okay people that are stopping at primary school in what way can we improve their skill we cannot steal them for them to live in meaningful to live in meaningful like we need three four years they have been empowered to make money for themselves i tell you this while we were young and majority of us did this job we worked in works here we worked in the kejab and we kejab industrial state to support ourselves after we finished secondary school and then in the university there's no amount of money your friends will give to you i did quickly a job i did it but if if i tell anyone that is watching if i tell the person that i put on that's the headband on my several times while i was still doing my course in uni that would they believe me so we must restore dignity in people and we must deemphasize this case for certain certificate and certificate and tertiary education empowerment there are technical skills that we need to go now unfortunately time is never on our side and never friend when we're talking about you know interesting issues but the conversation will never end or appreciate your time right here with us is it is a pleasure mercy and kofi it's a pleasure don't worry nijia and gana will come out of it absolutely um i think you've met you've used the word restructuring about a lost count of how many times you mentioned what even in education you use the word restructuring so i mean so gdm is a restructuring jobs and thank you very much for your time well do you have a fantastic all right then we need to move away and and that's because we would have to return for a first major conversation right here i'll be looking at what it means now that as was called the strike i mean we hear that it's conditionally but just before then let's tell you what happened today in history please stay with us


Well, hi there. I'm Sandy Allnock and today I'm going to watercolor this super, super sweet little bunny rabbit from Impression Obsession. I'm going to use one of the small sets of Daniel Smith Watercolors. This is the Skecher set to which I added a few other colors because I wanted to be able to use this set for things like this where I have maybe some animals in it. And this one has some interesting colors that I wanted to try together. And that's literally how I do a lot of my watercoloring is I try things together. So if you're looking to me to give you the exact colors you should use on something, then you're probably looking to me for more than I'm able to give you. I'm still new enough, even years into it at watercolor, that I don't really have an intuitive knowledge of what every color is going to do with every other color. So I'm just playing. And what I decided to do was use this Monty, yeah I can't even pronounce it, something something natural Sienna color with some bloodstone over top of it. Bloodstone is one of the Primitech colors and it's got a little bit of purple in it, a little bit of brown in it. The two of them together make a really interesting natural looking brown. And I'm putting them down in different mixtures. So some places it's more of one color, some places more of the other. And that's giving it a little bit more of a natural look as well. This is some Payne's Gray added into those colors and then a little bit more of the bloodstone. And just playing around with different quantities of different colors to see what kind of bunny rabbit that I could create out of the colors that are in this palette. These miniature palettes from Daniel Smith come with, as I've explained in previous videos, they come with six colors in the center row. There's center three colors and then the ones just below them are the ones that come with the set. And I've chosen all the other colors. So that is what this particular set is. I'll have all the colors listed on my blog if you're interested in knowing exactly which ones each one of them is. And there's a whole video that I did when I talked about each one of the sets and you can get more information on them. They're kind of a nice way to get started in a Daniel Smith palette without having to invest in getting a palette as well as the colors and the tubes and everything. You can get just a little set with six colors. And this one has two different red, yellow, blue combinations, which is nice. Now for the rest of my rabbit, I'm going to continue doing kind of the same thing except on his chest where I want it to feel like it's lighter fur, like it's white. White is never white. There's always some color in it. And you need to have some color to make it look like it's dimensional.  I'm not sure if this color will ever make it in my palette because I just can't say it well. But I'm using that very lightly with just a lot of water in it. And then just a very bare minimal touch of the blood stone because the blood stone might give it a little texture. And I don't know whether the texture is going to appear as it dries or whether it will appear at all in this quantity of pigment. Because the textures in a lot of the Primatec colors, they'll be really visible if you use them in particular ways. So if you use really thick paint, sometimes that granulation really pops out. And by granulation, I mean texture. And other times it's in certain colors when it's in combination with certain colors are used with water in particular ways. So any one of them you'd have to kind of play with them and see what brings out the most texture if that's what you're looking for. For his legs, I'm going to do the same thing that I did with the ears. Just paint a lot of that that natural Sienna color down on the legs themselves and then start adding other colors into it. And even though I'm kind of trying to make the amount of that Sienna color match here on the two legs, it really doesn't matter if you actually looked at a rabbit or any animal, you would see just like we have differences in our skin tones from one side of the face to the other, we have differences in the symmetry of our bodies and our features and that sort of thing from the left half to the right half of us, animals do too. And if they're a little bit different on one side of an animal than another, then so be it. The way that the pigment is moving is just blending on its own. Notice that I'm not mixing anything in a palette. So in order to watercolor, you don't have to mix in a palette. I actually prefer putting color directly on the paper and letting that be the mixing. And it's challenging sometimes because you have to move the pigment around and manage your quantities of pigment in a way that you can control a little bit of what happens. But I find it to be more interesting and spontaneous than constantly trying to mix up things in a palette in a large quantity. Now that makes it much more difficult for me to do entire scenes because I don't like to mix up tons of color for things, but you know, there you go. There's pluses and minuses to every way of painting. So my bunny now has a little pink nose and I've added a little bit of pink kind of trailing up into the nose. You can just barely see that. For the rabbit's back legs, now that the front legs are a little more dry, I'm going to do the same procedure, just adding that yellowish base color down first and then adding the other colors into it and making the color a little bit thicker and darker behind so that it starts to push the front legs forward and lets those back legs be in the back. And of course you can add more color to these and darken them as needed. If you want the color to blend in softly like this, then you have to do that while it's wet. And if you want to add texture onto it, like some of those little tiny strokes that I did, that you can do later once the paper is dry. And you'll notice that I'm not using a hairdryer or a heat gun or anything to dry my paper. I'm also not taping it down because what I find is if I can just adhere my paper in place and when I'm using, doing this with cards, I just use some ATG tape on the back to hold it down so it's not moving all over the table while I'm filming. But when I'm doing large pieces, I still don't tape anything down. There's sometimes when I do, if I want to have a specific border on it, if I want to have that white border on the edge of the paper. But what I have found is that if I heat set things, then the paper curls, because that's changing the way the paper functions. It's adding heat as an element to it and it may cause it all kinds of problems. So I find that it's safer to just let things air dry if I can. It also allows the paint to dry more naturally. The only time I use a hairdryer or a heat gun is if I have a particular area that is perfect as it is and I don't want it to move, one iota, then I might zap that particular spot because I want that spot to stay as is and not do any bleeding or develop any blooms or anything like that. So while the rest of this is drying, I'm going to start painting the flowers on the top. The flowers are in set. There's a couple different types of flower clumps and this one seemed to be a good crown for the little bunny. I even stamped it so that it kind of comes over one eye. I thought that looked a little bit more playful than it might otherwise and masked that portion out to just stamp the bunny so he kind of appears up and under there. My masking is not always perfect so I did end up coloring that right or the his left eye, the right eye looking at the screen. I did have to fill that back in because my mask wasn't perfect but I just did that with a waterproof pen like a Sharpie and you could fix little areas like that and hello to the puppies barking outside. I don't know if you heard that but I know some of you guys noticed that there's little noises in the back of my videos from time to time because I get busy trying to get my voiceovers done and the kids are playing. Oh the joys of working at home when you have dogs especially young dogs who love to wrestle and play a whole lot. So I'm using a couple of these different colors to create some flowers on the top painting around them with pinks and yellows and greens and that sort of thing just having kind of some fun with that and letting that be very flat color. I'm not trying to put in a whole lot of shading just a little tiny bit of darker color in in one side of it for a little bit of shadows but I really loved how the bunny came out and I didn't want to risk doing anything funky to it. I did want to add a little bit of background and not a ton. I don't want it to feel like it's a detailed scene or anything but the bunny needed a little bit of home to be in somehow so I've created a horizon line behind there and I'm just going to paint one color this serpentine green going upward and as I get to the top section just going to use some dry brush to create almost just some little texture that feels like a plant and when I got that part done I thought oh what if I added a little bit of one of the colors that's in this palette that that I like quite a bit is nickel azo and so I was thinking as I was doing this do I add some nickel azo to this and see what happens and so I tried that once I got my my base serpentine green down there and what I found is that my paint was not wet enough to actually do anything so I ended up with these blobs of the nickel azo instead of that wonderful burst that it tends to do when you drop it into another color and I'll have to play with it and see if it's the fact that it's serpentine green or if it's the fact that it was very dry because I did use that dry brush on the top which is where I wanted that bright yellow so then I decided after that that I would make it look a little rougher overall by adding some of my darker green which in this particular case is under sea green and just adding a little bit of that to create a little bit of dimension a little bit of texture within that green yellow blobby area it's just going to disappear in the background it's going to dry lighter than this so I just wanted something back there that's going to give it a setting rather than worrying too much about trying to make it look like anything which I think it was successful I mounted it onto a nice heavy car base and called it done and what I decided to do was to put the sentiment on the inside because this sentiment is quite hilarious I'm totally not sure for what occasion I will send this card I don't really know it might just be one of those for my patrons because they like fun cards regardless but who knows we'll see it was a fun card to make it was really fun to paint and I'm going to be coloring this bunny a whole lot it's one of those stamps that's probably not going to make it into the prize box for quite some time because he's really fun to color in a lot of different mediums I will see you guys later have a great day