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Chapter 4 the Lebesgue Spaces

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Chapter 4 the Lebesgue Spaces Chapter 4 The Lebesgue Spaces In this chapter we study Lp-integrable functions as a function space. Knowledge on functional analysis required for our study is briefly reviewed in the first two sections. In Section 1 the notions of normed and inner product spaces and their properties such as completeness, separability, the Heine-Borel property and espe- cially the so-called projection property are discussed. Section 2 is concerned with bounded linear functionals and the dual space of a normed space. The Lp-space is introduced in Section 3, where its completeness and various density assertions by simple or continuous functions are covered. The dual space of the Lp-space is determined in Section 4 where the key notion of uniform convexity is introduced and established for the Lp-spaces. Finally, we study strong and weak conver- gence of Lp-sequences respectively in Sections 5 and 6. Both are important for applications. 4.1 Normed Spaces In this and the next section we review essential elements of functional analysis that are relevant to our study of the Lp-spaces. You may look up any book on functional analysis or my notes on this subject attached in this webpage. Let X be a vector space over R. A norm on X is a map from X ! [0; 1) satisfying the following three \axioms": For 8x; y; z 2 X, (i) kxk ≥ 0 and is equal to 0 if and only if x = 0; (ii) kαxk = jαj kxk, 8α 2 R; and (iii) kx + yk ≤ kxk + kyk. The pair (X; k·k) is called a normed vector space or normed space for short. The norm induces a metric on X given by d(x; y) = kx − yk : 1 Hence, a normed space is always a metric space and a topological one. We can talk about convergence, continuity, etc. in a normed space. The notion of a Cauchy sequence also makes sense in a metric space, that is, a sequence fxng in a metric space (X; d) is called a Cauchy sequence if for every " > 0, there is an n0 such that d(xn; xm) < "; for all n; m ≥ n0. Recall that a metric is complete if every Cauchy sequence is convergent. A complete normed space is called a Banach space. A general result asserts that every normed space is a dense subspace of some Banach space, and this Banach space is called the completion of the normed space. Here are some examples of Banach spaces. n qP 2 • R is a Banach space under the Euclidean norm kxk = xj . p n P p o • ` = x = (x1; x2;::: ): j jxjj < 1; xj 2 R , 1 ≤ p < 1; under the 1=p p P p ` -norm: kxkp = j jxjj . • C([0; 1]) = fcontinuous functions on [0; 1]g under the uniform norm (or sup-norm) kfk1 = sup fjf(x)j : x 2 [0; 1]g = max fjf(x)j : x 2 [0; 1]g : • C(K) where K is a compact subset in Rn or a compact metric space. It is also a normed space under the sup-norm. It becomes C([0; 1]) when K = [0; 1]. Let X1 be a subspace of the normed space (X; k·k). Then (X1; k·k) is again a normed space. The followings are some subspaces in C([0; 1]): P ([0; 1]) = fThe restrictions of all polynomials on [0; 1]g ; C1([0; 1]) = fAll continuously differentiable functions on [0; 1]g ; C0([0; 1]) = fAll continuous functions with f(0) = f(1) = 0g ; Cc((0; 1)) = fAll continuous functions on (0; 1) vanishing near 0 and 1g : Among these four subspaces, only the third one is a Banach space. The properties of finite dimensional and infinite dimensional normed spaces are very different. Let us look at the finite dimensional one first. The Euclidean space (Rn; k·k) is a typical finite dimensional normed space. It has the following four basic properties. Later we will contrast them with those in infinite dimensional spaces. • Completeness. Every finite dimensional normed space is a Banach space. 2 • Separability. Recall that a topological space is separable if it has a count- able, dense subset. Rn is separable as it possesses the countable dense subset Qn. • The Heine-Borel Property. A normed space has the Heine-Borel property if every bounded sequence in X has a convergent subsequence. It is well- known that Rn has this property. • Projection Property. Let X1 be a proper, closed subspace of the normed space X and x0 not in X1. X is called to have the projection property if there exists some z 2 X1 such that kz − x0k = inf fkx − x0k : x 2 X1g : In other words, there is a point on X1 realizing the distance from x0 to X1. Rn has this property. In fact, it is instructive to see how this is proved. Let fxjg be a minimizing sequence of d, i.e., kxj − x0k ! d as j ! 1. Taking " = 1, there is some j0 such that kxj − x0k ≤ d + 1, 8j ≥ j0. It follows that fxjg is a bounded sequence: kxjk ≤ kxj − x0k + kx0k ≤ d + 1 + kx0k. By the Heine-Borel property, there is a convergent subsequence fxjk g, xjk ! z for some z 2 X1. By the continuity of the norm, kxjk − x0k ! kz − x0k. On the other hand, kxjk − x0k ! d as fxjg is minimizing. We conclude that kz − x0k = d. An inner product on the normed space X over R is a map : X × X ! R satisfying, for all x; y; z 2 X, (i) hx; xi ≥ 0, and equal to 0 if and only if x = 0, (ii) hx; yi = hy; xi, (iii) hαx + βy; zi = α hx; zi + β hy; zi, for all α; β 2 R, Note it follows that hx; αy + βzi = α hx; yi + β hx; zi : The pair (X; h ; i) is called in inner product space. The Euclidean product, or the dot product, on Rn is given by n X hx; yie = xjyj: j=1 It makes Rn an inner product space. In general, an inner product induces a norm by q kxk = hx; xie: 3 For instance, the Euclidean norm comes from the Euclidean product. An inner product space is called a Hilbert space if it is a Banach space in the induced norm. Using the inner product structure on the Euclidean space, we have the follow- ing characterization of the point that minimizes the distance between x0 and X1 in the projection property, namely, z is the projection of x0 onto X1 if and only it satisfies hx; x0 − zi = 0; 8x 2 X1: (4.1) Now I compare C([0; 1]), or C[0; 1] for simplicity, with the Euclidean space. It is easy to show that C[0; 1] is an infinite dimensional vector space. • Completeness. Yes, C[0; 1] is complete under the sup-norm. In elemen- tary analysis we proved that the uniform limit of a sequence of continuous functions is continuous. • Separability. Yes. By the Weierstrass approximation theorem, every contin- uous function on [0; 1] can be approximated by polynomials with rational coefficients. Thus the collection of all these polynomials forms a count- able, dense set in C([0; 1]). More generally, it can be shown that C(K) is separable when K is a compact metric space. • The Heine-Borel Property. Not any more. In fact, let fn be the piecewise linear function which is equal to 0 at 0 and [1=n; 1] and fn(1=2n) = 1. Clearly fn(x) ! 0 8x 2 [0; 1]. However, it does not contain any uniformly convergent subsequence. Suppose on the contrary that there exists some ffnj g converging to some continuous f. Then f must be the zero function. For every " > 0, there is some j0 such that fnj − 0 1 < "; j ≥ j0: Taking " = 1=2, fnj 1 < 1=2 for all large j. But fnj 1 = 1, contradic- tion holds. Hence the Heine-Borel property no longer holds in C([0; 1]). It is a bit striking that this property characterize finite dimensionality. Theorem 4.1. The Heine-Borel property holds on a normed space X if and only if X is finite dimensional. See, for instance, Theorem 2.1.2 in my notes on functional analysis. • Projection Property. No. But the construction of an example is not easy. I just want to point out that very often a property so obvious for a finite dimensional space may not hold in an infinite dimensional space. If it holds, more effort is need to established it. Fortunately, this property holds in Hilbert spaces. 4 Theorem 4.2. Let X be a Hilbert space and x0 2= X1 where X1 is a proper closed subspace. There exists a unique z 2 X1 such that kz − x0k = inf fkx − x0k : x 2 X1g ; and (4.1) holds. Proof. First of all, by expanding kx ± yk2= hx ± y; x ± yi, we have the parallel- ogram rule kx + yk2+kx − yk2= 2(kxk2+kyk2) which is true on every inner product space. Now, let fykg be a minimizing sequence for d = inf fky − x0k : y 2 X1g, that is, kyk − x0k! d as k ! 1: Using the parallelogram rule, we have y + y ky − y k2= 2(ky − x k2+ky − x k2) − 2k k m − x k2: k m k 0 m 0 2 0 As (yk + ym)=2 belongs to X1, k(yk + ym)=2 − x0k≥ d: Given " > 0, there exists 2 2 some n0 such that kyn − x0k −d < " for all n ≥ n0. It follows that 2 2 kyk − ymk≤ 2d + 2" − 2d = 2"; for all k; m ≥ n0.
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