## Linear Operators: General theory |

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Page 8

Hence , by Theorem 7 , there is a maximal well - ordered

for if x is in E but not in E , the ordering so in ... 2 If a family & of

has the property that A CE if and only if every finite

...

Hence , by Theorem 7 , there is a maximal well - ordered

**subset**E. of E. Now E. Efor if x is in E but not in E , the ordering so in ... 2 If a family & of

**subsets**of a sethas the property that A CE if and only if every finite

**subset**of A belongs to & , then...

Page 439

Let K be a

A CK is said to be an extremal

1 - a ) ka , 0 ) < a < 1 , of two points of K is in A only if both k , and ky are in A. An ...

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 akı + (1 - a ) ka , 0 ) < a < 1 , of two points of K is in A only if both k , and ky are in A. An ...

Page 456

( Schauder - Tychonoff ) A compact convex

topological space has the fixed point property . Proof . By Zorn's lemma there

exists a minimal convex

previous ...

( Schauder - Tychonoff ) A compact convex

**subset**of a locally convex lineartopological space has the fixed point property . Proof . By Zorn's lemma there

exists a minimal convex

**subset**K of K with the property that TK , CK ;. By theprevious ...

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero