## Linear Operators: General theory |

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Page 439

Let K be a

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

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

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 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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### Common terms and phrases

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