## 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 318

Vector

last section permit us to develop a more satisfactory theory of vector

countably additive set functions ( briefly , vector

able ...

Vector

**Valued**Measures The theorems on spaces of set functions proved in thelast section permit us to develop a more satisfactory theory of vector

**valued**countably additive set functions ( briefly , vector

**valued**measures ) than we wereable ...

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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

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