## 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 | |

87 other sections not shown

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### Common terms and phrases

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