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Page 439
Let K be a subset of a real or complex linear vector space X. A non - void subset A CK is said to be an extremal subset of K if a proper convex combination ak ; + ( 1 - a ) ką , 0 < a < 1 , of two points of K is in A only if both k ...
Let K be a subset of a real or complex linear vector space X. A non - void subset A CK is said to be an extremal subset of K if a proper convex combination ak ; + ( 1 - a ) ką , 0 < a < 1 , of two points of K is in A only if both k ...
Page 440
totally ordered subfamily of A , the non - void set n A , is a closed extremal subset of K which furnishes a lower bound for A 1 . It follows by Zorn's lemma that A contains a minimal element Ao . Suppose that A , contains two distinct ...
totally ordered subfamily of A , the non - void set n A , is a closed extremal subset of K which furnishes a lower bound for A 1 . It follows by Zorn's lemma that A contains a minimal element Ao . Suppose that A , contains two distinct ...
Page 459
12 Let X be a B - space , and let K be a weakly compact convex subset of X. Show that K has a continuous tangent functional at each point of a dense subset of its boundary . 13 Let X be a B - space , and let K * be a bounded X - closed ...
12 Let X be a B - space , and let K be a weakly compact convex subset of X. Show that K has a continuous tangent functional at each point of a dense subset of its boundary . 13 Let X be a B - space , and let K * be a bounded X - closed ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero