## Linear Operators: General theory |

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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 ; ) = u ( E ; ) i = 1 - 1 whenever E2 , E2 , . . . are disjoint sets in E ...

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 ; ) = u ( E ; ) i = 1 - 1 whenever E2 , E2 , . . . are disjoint sets in E ...

Page 323

A scalar

sequence { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -

almost everywhere ; ( ii ) the sequence { Seln ( 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 { n } of simple functions such that ( i ) In ( s ) converges to f ( s ) 4 -

almost everywhere ; ( ii ) the sequence { Seln ( 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 | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero