## Linear Operators: General theory |

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Page 95

Finitely Additive Set

process is defined with respect to a scalar

integral v ...

Finitely Additive Set

**Functions**In contrast to the terms real**function**, complex**function**, etc . , where the adjectives real and ... Thus , even if the integrationprocess is defined with respect to a scalar

**valued**set**function**u , the resultingintegral v ...

Page 119

( b ) the

, is u - measurable . Next suppose that we consider a

extended real -

NC S ...

( b ) the

**function**g defined by g ( s ) = f ( s ) if s ¢S + US - , g ( s ) = 0 if s e S + US -, is u - measurable . Next suppose that we consider a

**function**† ( vector orextended real -

**valued**) which is defined only on the complement of a u - null setNC S ...

Page 196

Next we study the relation between the theory of product measures and the

theory of vector

and F is a u - measurable

Next we study the relation between the theory of product measures and the

theory of vector

**valued**integrals . ... Suppose that ( S , E , u ) is a measure spaceand F is a u - measurable

**function**whose values are in L ( T , ET , 2 ) , 1 < p < oo .### What people are saying - Write a review

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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 analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero