## Linear Operators: General theory |

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Page 95

A set function is a function defined on a family of sets , and having values either

in a B - space , which may be the set of real or complex numbers , or in the

the ...

A set function is a function defined on a family of sets , and having values either

in a B - space , which may be the set of real or complex numbers , or in the

**extended**real number system , in which case its range contains at most one ofthe ...

Page 126

In this case the results of the preceding sections can be considerably

1 DEFINITION . Let u be a vector valued , complex valued , or

valued additive set function defined on a field of subsets of a set S . Then u is

said to ...

In this case the results of the preceding sections can be considerably

**extended**.1 DEFINITION . Let u be a vector valued , complex valued , or

**extended**realvalued additive set function defined on a field of subsets of a set S . Then u is

said to ...

Page 137

The total variation of a regular additive complex or

function on a field is regular . Moreover , the positive and negative variations of a

bounded regular real valued additive set function are also regular . Proof . If u is a

...

The total variation of a regular additive complex or

**extended**real valued setfunction on a field is regular . Moreover , the positive and negative variations of a

bounded regular real valued additive set function are also regular . Proof . If u is a

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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 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