## Linear Operators: General theory |

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Page 21

A set is said to be

A set is said to be

**dense**in a topological space X , if its closure is X. It is said to be nowhere**dense**if its closure does not contain any open set . A space is separable , if it contains a denumerable**dense**set . 12 THEOREM .Page 450

If a convex subset of a separable B - space X has an interior point , it has a unique tangent at each point of a

If a convex subset of a separable B - space X has an interior point , it has a unique tangent at each point of a

**dense**subset of its boundary . PROOF . Let K be the convex set . It will be shown that — ( x , y ) : T ( x , -y ) , y e X ...Page 842

... IV.9.7 ( 308 ) Yosida - Hewitt decomposition , ( 233 ) De Morgan , rules of , ( 2 )

... IV.9.7 ( 308 ) Yosida - Hewitt decomposition , ( 233 ) De Morgan , rules of , ( 2 )

**Dense**convex sets , V.7.27 ( 437 )**Dense**linear manifolds , 1.7.10-41 ( 438-439 )**Dense**set , definition , 1.6.11 ( 21 ) density of simple functions ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian 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