## Linear Operators: General theory |

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Page 410

A set K CX is

A set K CX is

**convex**if x , y € K , and 0 sa si , imply ax + ( 1 - aby e K. The following lemma is an obvious consequence of Definition 1 . 2 LEMMA . The intersection of an arbitrary family of**convex**subsets of the linear space X is ...Page 418

If K , K , are disjoint closed ,

If K , K , are disjoint closed ,

**convex**subsets of a locally**convex**linear topological space X , and if Ky is compact , then some non - zero continuous linear functional on X separates K , and Ką . 12 COROLLARY .Page 461

and an arbitrary

and an arbitrary

**convex**set is possible , provided they are disjoint ( compare Theorem 2.8 ) . He also proved that a**convex**set K which is compact in the X * topology of the normed linear space X , can be separated from an arbitrary ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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