## Linear Operators, Part 1 |

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Page 95

In some cases that will be encountered the values of u are not scalars , but

customarily where integration is used in this text u is a scalar

f a vector ( or scalar )

defined ...

In some cases that will be encountered the values of u are not scalars , but

customarily where integration is used in this text u is a scalar

**valued**function andf a vector ( or scalar )

**valued**function . Thus , even if the integration process isdefined ...

Page 126

Let u be a vector

function defined on a field of subsets of a set S . Then u is said to be countably

additive if u ( UE ; ) = Eu ( E ; ) i = 1 i = 1 whenever E1 , E2 , . . . are disjoint sets in

...

Let u be a vector

**valued**, complex**valued**, or extended real**valued**additive setfunction defined on a field of subsets of a set S . Then u is said to be countably

additive if u ( UE ; ) = Eu ( E ; ) i = 1 i = 1 whenever E1 , E2 , . . . are disjoint sets in

...

Page 323

A scalar

sequence { In } of simple functions such that ( i ) fr ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Setn ( s ) u ( ds ) } converges in the norm

of X for ...

A scalar

**valued**measurable function f is said to be integrable if there exists asequence { In } of simple functions such that ( i ) fr ( s ) converges to f ( s ) u -

almost everywhere ; ( ii ) the sequence { Setn ( s ) u ( ds ) } converges in the norm

of X for ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero