## Linear Operators, Part 1 |

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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 E whose interior contains E such that \ u ( C ) | < ε ...

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

is said to be

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

Page 170

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

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 in ...

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

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 in ...

Page 853

( See Reflexivity )

170 ) ...

( 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.9.19-22 (170 ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### 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

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