## 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 451

We wish to prove that Z = n = 2n = nn - Zn , i is

We wish to prove that Z = n = 2n = nn - Zn , i is

**dense**in X. Suppose that Z is not**dense**in X , and that p ¢ 2. Then some sphere S ( p , ε ) does not intersect Z. If S = S ( p , ε / 2 ) , then SZ = d . Hence U USZ = S. It follows from ...Page 842

... rules of , ( 2 )

... rules of , ( 2 )

**Dense**convex sets , V.7.27 ( 437 )**Dense**linear manifolds , V.7.10-1 ( 438-439 )**Dense**set , definition , 1.6.11 ( 21 ) density of simple functions in L , isp < 0 , III.3.8 ( 125 ) density of continuous functions in ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

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