## Linear Operators: General theory |

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Page 21

A set is said to be

be nowhere

separable , if it contains a denumerable

topological ...

A set is said to be

**dense**in a topological space X , if its closure is X . It is said tobe nowhere

**dense**if its closure does not contain any open set . A space isseparable , if it contains a denumerable

**dense**set . 12 THEOREM . If atopological ...

Page 451

If we put Zn , i = V - Zn , i , j , then 2m , i is open in X , and Zn = nie zmi We wish to

prove that Z = nu Zn = nomine Zni is

, and that p¢ Ž . Then some sphere S ( p , a ) does not intersect Z . If S = S ( p ...

If we put Zn , i = V - Zn , i , j , then 2m , i is open in X , and Zn = nie zmi We wish to

prove that Z = nu Zn = nomine Zni is

**dense**in X . Suppose that Z is not**dense**in X, and that p¢ Ž . Then some sphere S ( p , a ) does not intersect Z . If S = S ( p ...

Page 842

7 ( 128 ) , III . 4 . 11 ( 130 ) Lebesgue decomposition , III . 4 . 14 ( 132 ) Saks

decomposition , IV . 9 . 7 ( 308 ) Yosida - Hewitt decomposition , ( 233 ) De

Morgan , rules of , ( 2 )

manifolds , V . 7 .

7 ( 128 ) , III . 4 . 11 ( 130 ) Lebesgue decomposition , III . 4 . 14 ( 132 ) Saks

decomposition , IV . 9 . 7 ( 308 ) Yosida - Hewitt decomposition , ( 233 ) De

Morgan , rules of , ( 2 )

**Dense**convex sets , V . 7 . 27 ( 437 )**Dense**linearmanifolds , V . 7 .

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero