## Linear Operators: General theory |

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Page 439

Extremal Points 1 DEFINITION . Let K be a

vector space X . A non - void

a proper convex combination ak , + ( 1 - a ) k , 0 < a < 1 , of two points of K is in A

only ...

Extremal Points 1 DEFINITION . Let K be a

**subset**of a real or complex linearvector space X . A non - void

**subset**A CK is said to be an extremal**subset**of K ifa proper convex combination ak , + ( 1 - a ) k , 0 < a < 1 , of two points of K is in A

only ...

Page 440

totally ordered subfamily of A , the non - void set n A , is a closed extremal

of K which furnishes a lower bound for A 1 . It follows by Zorn ' s lemma that A

contains a minimal element A . Suppose that A , contains two distinct points p and

...

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 A . Suppose that A , contains two distinct points p and

...

Page 459

12 Let X be a B - space , and let K be a weakly compact convex

Show that K has a continuous tangent functional at each point of a dense

of its boundary . 13 Let X be a B - space , and let K * be a bounded X - closed

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

convex ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero