## 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 ll - measurable . Next suppose that we consider a

extended real -

NCS .

( b ) the

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

**function**í ( vector orextended real -

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

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 sp < 0o .### What people are saying - Write a review

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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