## Linear Operators, Part 1 |

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Page 137

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

is said to be

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

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

is said to be

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

Page 170

17 Suppose that S is a normal topological space and that u is

defined on the field of Borel sets in S . Show that if X is separable , the set of

continuous functions in TM ( S , E , u , X ) is dense in TM ( S , E , M , X ) . Show

that for 1 < p ...

17 Suppose that S is a normal topological space and that u is

**regular**anddefined on the field of Borel sets in S . Show that if X is separable , the set of

continuous functions in TM ( S , E , u , X ) is dense in TM ( S , E , M , X ) . Show

that for 1 < p ...

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

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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