## Linear Operators: General theory |

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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 –1 ( 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 , V.7.40-41 ( 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

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