Theorem 118 the unit ball of the dual of a normed

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Theorem 118. The unit ball of the dual of a normed space X is compact in the weak star topology. Our proof of the Riesz representation theorem used the Hahn-Banach theorem as a convenience. Our proof of the next result uses it as basic ingredient. Theorem 119. Every Banach space is isometrically isomorphic to some subspace of C ( K ) for some compact space K . (In my opinion this result looks more interesting than it is.) Our third result requires us to recast the Hahn Banach theorem in a geometric form. Lemma 120. If V is a real normed spaced and E is a convex subset of V containing B ( 0 , ² ) for some ² > 0 , then, given any x / E we can find a continuous linear map T : V R such that T x T e for all e E . 30
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Theorem 121. If V is a real normed spaced and K is a compact convex subset of V , then, given any x / E we can find a continuous linear map T : V R and a real α such that T x > α > T k for all k K . Definition 122. Let V be a real or complex vector space. If K is a non- empty subset of V we say that E K is an extreme set of K if, whenever u, v K , 1 > λ > 0 and λu + (1 - λ ) v E , it follows that u, v E . If { e } is an extreme set we call e an extreme point. Exercise 123. Define an extreme point directly. Exercise 124. We work in R 2 . Find the extreme points, if any, of the following sets and prove your statements. (i) E 1 = { x : k x k < 1 } . (ii) E 2 = { x : k x k ≤ 1 } . (iii) E 3 = { ( x, 0) : x R } . (iv) E 4 = { ( x, y ) : | x | , | y | ≤ 1 } . Theorem 125. (Krein-Milman). A non-empty compact convex subset K of a normed vector space has at least one extreme point. Theorem 126. A non-empty compact convex subset K of a normed vector space is the closed convex hull of its extreme points (that is, is the smallest closed convex set containing its extreme points). Our hypotheses in our version of the Krein-Milman theorem are so strong as to make the conclusion practically useless. However the hypotheses can be much weakened as is indicated by the following version. Theorem 127. (Krein-Milman). Let E be the dual space of a normed vector space. A non-empty convex subset K which is compact in the weak star topology has at least one extreme point. Theorem 128. Let E be the dual space of a normed vector space. A non- empty convex subset K which is compact in the weak star topology is the weak star closed convex hull of its extreme points. Lemma 129. The extreme points of the closed unit ball of the dual of C ([0 , 1]) are the delta masses δ a and - δ a with a [0 , 1] . 14 The Rivlin-Shapiro formula In this section we give an elegant use of extreme points due to Rivlin and Shapiro. 31
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Lemma 130. Carath´ eodory We work in R n . Suppose that x R n and we are given a finite set of points e 1 , e 2 , . . . , e N and positive real numbers λ 1 , λ 2 , . . . , λ N such that N X j =1 λ j = 1 , N X j =1 λ j e j = x . Then after renumbering the e j we can find positive real numbers λ 0 1 , λ 0 2 , . . . , λ 0 m with m n + 1 such that m X j =1 λ 0 j = 1 , m X j =1 λ 0 j e j = x .
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