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

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

of X ...

A scalar

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

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

of X ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero