## Linear Operators: General theory |

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Page 100

Let u be an additive set function defined on a field of subsets of a set S . A subset

N of S is said to be a u -

10 that every subset of a u -

...

Let u be an additive set function defined on a field of subsets of a set S . A subset

N of S is said to be a u -

**null set**if v * ( u , N ) ... It follows immediately from Lemma10 that every subset of a u -

**null set**, and every finite union of u -**null sets**, is a u...

Page 147

A set is a

u , F ) = 0 . PROOF . If E is a

measurable sets En containing E with v ( u , En ) < 1 / n . Thus the set F = n E , is

a ...

A set is a

**null set**if and only if it is a subset of some measurable set F such that v (u , F ) = 0 . PROOF . If E is a

**null set**, then v * ( u , E ) = 0 and there aremeasurable sets En containing E with v ( u , En ) < 1 / n . Thus the set F = n E , is

a ...

Page 213

( a ) If for each p in a set ACG 2 ( C ) lim inf . < ? ' , H ( C ) - 0 u ( C ) where C is a

closed cube containing p , then each neighborhood of A contains an open set Q

such that A - Q is a d -

...

( a ) If for each p in a set ACG 2 ( C ) lim inf . < ? ' , H ( C ) - 0 u ( C ) where C is a

closed cube containing p , then each neighborhood of A contains an open set Q

such that A - Q is a d -

**null set**and 1 ( Q ) < ru ( Q ) . ( b ) If for each p in a set A CG...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero