## 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 -t ( x , y ) T ( x , -y ) , y e X ...Page 451

We wish to prove that Z = n -- Zn novini Zn , is

We wish to prove that Z = n -- Zn novini Zn , is

**dense**in X. Suppose that Z is not**dense**in X , and that p ¢ Ž . Then some sphere S ( p , e ) does not intersect Z. If S = S ( p , 8/2 ) , then SZ = $ . Hence U USZ , = S. It follows from ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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