## Linear Operators: General theory |

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Page 106

Nelson Dunford, Jacob T. Schwartz. 10 DEFINITION . The functions totally

functions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E

in E with v ...

Nelson Dunford, Jacob T. Schwartz. 10 DEFINITION . The functions totally

**u**-**measurable**on S , or , if u is understood , totally measurable on S are thefunctions in the closure TM ( S ) in F ( S ) of the u - simple functions . If for every E

in E with v ...

Page 119

( b ) the function g defined by g ( s ) = f ( s ) if s ¢ S + US - , g ( s ) = 0 if se S + US -

, is

extended real - valued ) which is defined only on the complement of a u - null set

NCS .

( b ) the function g defined by g ( s ) = f ( s ) if s ¢ S + US - , g ( s ) = 0 if se S + US -

, is

**u**-**measurable**. Next suppose that we consider a function f ( vector orextended real - valued ) which is defined only on the complement of a u - null set

NCS .

Page 178

But old , E ) = 1 , \ | ( s ) [ 0 ( u , ds ) so that f ( s ) = 0 for s in E except in a set A with

v ( u , A ) = 0 . Thus 8n ( $ ) { ( 8 ) ▻g ( s ) [ ( 8 ) , 8€ F - A , and Corollary 6 . 14

shows that Xefg is

...

But old , E ) = 1 , \ | ( s ) [ 0 ( u , ds ) so that f ( s ) = 0 for s in E except in a set A with

v ( u , A ) = 0 . Thus 8n ( $ ) { ( 8 ) ▻g ( s ) [ ( 8 ) , 8€ F - A , and Corollary 6 . 14

shows that Xefg is

**u**-**measurable**. Conversely , let fg be**u**-**measurable**, and let...

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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