## Linear Operators, Part 1 |

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Page 137

An additive set function u defined on a

is said to be regular if for each Ec { and ε > 0 there is a set F in E whose closure is

contained in E and a set

An additive set function u defined on a

**field**? of subsets of a topological space Sis said to be regular if for each Ec { and ε > 0 there is a set F in E whose closure is

contained in E and a set

**G**in E whose interior contains E such that \ u ( C ) | < ε ...Page 166

Suppose that E is a set in E. If we put E ( E ) = { F € E | F CE } it is clear that E ( E )

is a

measure on E. Similarly , if the domains of two functions f ,

E ...

Suppose that E is a set in E. If we put E ( E ) = { F € E | F CE } it is clear that E ( E )

is a

**field**of subsets of E , and that E ... defined on E is said to converge in u -measure on E. Similarly , if the domains of two functions f ,

**g**both contain the setE ...

Page 168

Q.E.D. 5 LEMMA . Let ( S , E , u ) be a positive measure space and

subset of L , ( S , E , u , X ) , where 1 p < .0 . Then there is a set s , in E , a sub o -

...

Q.E.D. 5 LEMMA . Let ( S , E , u ) be a positive measure space and

**G**a separablesubset of L , ( S , E , u , X ) , where 1 p < .0 . Then there is a set s , in E , a sub o -

**field**of E ( S ) , and a closed separable subspace X1 of 2 such that the restriction...

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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