## Linear Operators, Part 1 |

### From inside the book

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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 224

It will be assumed that the reader is familiar with the elementary theory of

complex

applied here to obtain the extensions which will be used later . Throughout this

section ...

It will be assumed that the reader is familiar with the elementary theory of

complex

**valued**analytic functions of one complex variable and this theory will beapplied here to obtain the extensions which will be used later . Throughout this

section ...

Page 323

A scalar

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

almost everywhere ; ( ii ) the sequence { Sein ( 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 ) Ín ( s ) converges to f ( s ) u -

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

of X for ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero