## Linear Operators: General theory |

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Page 137

An additive set function u defined on a field of subsets of a topological space S is

said to be

contained in E and a set G in whose interior contains E such that lu ( C ) < ε for ...

An additive set function u defined on a field of subsets of a topological space S is

said to be

**regular**if for each Ee and a > 0 there is a set F in whose closure iscontained in E and a set G in whose interior contains E such that lu ( C ) < ε for ...

Page 170

16 Show that E forms an algebra in which A2 A if we define AB ANB , A + B = A

AB and that the u - null sets are an ideal in this algebra . 17 Suppose that S is a

normal topological space and that u is

sets ...

16 Show that E forms an algebra in which A2 A if we define AB ANB , A + B = A

AB and that the u - null sets are an ideal in this algebra . 17 Suppose that S is a

normal topological space and that u is

**regular**and defined on the field of Borelsets ...

Page 853

( See Reflexivity )

( See Reflexivity )

**Regular**closure , ( 462 - 463 )**Regular**convexity , ( 462 – 463 )**Regular**element in a ring , ( 40 )**Regular**method of summability , II . 4 . 35 ( 75 )**Regular**set function . ( See also Set function ) additional properties , III . 9 .### What people are saying - Write a review

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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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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