## Linear Operators: General theory |

### From inside the book

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Page 136

There is a

o - field containing a given family of sets . ... additive non - negative extension to

the o - field determined by E . If u is o - finite on then this extension is

There is a

**uniquely**determined smallest field and a**uniquely**determined smallesto - field containing a given family of sets . ... additive non - negative extension to

the o - field determined by E . If u is o - finite on then this extension is

**unique**.Page 202

We will first show that u is

with the stated value on elementary sets . For each r let Mos han be set functions

on En defined by the formulas M7 ( Ex ) = u ( EnXS . . ) , ( Ex ) = 2 ( E , SL ) , E ,

CE ...

We will first show that u is

**unique**. Let 2 be another additive set function on £ ,with the stated value on elementary sets . For each r let Mos han be set functions

on En defined by the formulas M7 ( Ex ) = u ( EnXS . . ) , ( Ex ) = 2 ( E , SL ) , E ,

CE ...

Page 516

42 Show that in Exercise 38 the set function u is

factor if and only if n - 1 & n = 1 / ( $ ' ( s ) ) converges uniformly to a constant for

each fe B ( S ) . 43 Show that in Exercise 39 the measure u is

42 Show that in Exercise 38 the set function u is

**unique**up to a positive constantfactor if and only if n - 1 & n = 1 / ( $ ' ( s ) ) converges uniformly to a constant for

each fe B ( S ) . 43 Show that in Exercise 39 the measure u is

**unique**up to a ...### 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 | |

50 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

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