## Linear Operators: General theory |

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Page 97

The total variation v ( u ) of an

The total variation v ( u ) of an

**additive**set function u is important because it dominates u in the sense that v ( u , E ) 2 \ u ( E ) for E € £ ; the reader should test his comprehension of Definition 4 below by proving that v ( u ) ...Page 126

Countably

Countably

**Additive**Set Functions The basis for the present section is a countably**additive**set function defined on a o - field of subsets of a set . In this case the results of the preceding sections can be considerably extended .Page 233

All of these integrals are countably

All of these integrals are countably

**additive**integrals . In Section IV.10 we will present a Lebesgue type theory of integration of a scalar function with respect to a countably**additive**vector valued measure .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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