## 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 , ; , then Zn , is open in X , and Zn = n Zini We wish to

prove that Z = n Zn = nemocnice Zn , is

in X , and that p¢Z . Then some sphere S ( p , e ) does not intersect Z . If S = S ( p ,

ε ...

If we put Zn , i = V - Zn , i , ; , then Zn , is open in X , and Zn = n Zini We wish to

prove that Z = n Zn = nemocnice Zn , is

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

ε ...

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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

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