## Linear Operators, Part 1 |

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Page 95

Finitely Additive Set

process is

integral v ...

Finitely Additive Set

**Functions**In contrast to the terms real**function**, complex**function**, etc . , where the adjectives real and ... Thus , even if the integrationprocess is

**defined**with respect to a scalar valued set**function**u , the resultingintegral v ...

Page 98

The total variation of an additive set

S is also additive on E . Proof . Let { A ; } be a finite sequence of disjoint sets in E

with ACEUF where E , F € £ and EF = $ . Let E ; = EA , F ; = FA ; , then Elu ( A ...

The total variation of an additive set

**function defined**on a field of subsets of a setS is also additive on E . Proof . Let { A ; } be a finite sequence of disjoint sets in E

with ACEUF where E , F € £ and EF = $ . Let E ; = EA , F ; = FA ; , then Elu ( A ...

Page 101

SES SES If for some null set N the restriction of f to $ - N has its values in a

separable set , the

Expressions such as essentially compact valued are

example above ...

SES SES If for some null set N the restriction of f to $ - N has its values in a

separable set , the

**function**f is said to be il - essentially separably valued .Expressions such as essentially compact valued are

**defined**similarly . In theexample above ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 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 obtained 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