## Linear Operators: General theory |

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Page 98

The total variation of an additive set

The total variation of an additive set

**function defined**on a field of subsets of a 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 = $ .Page 101

SES SES If for some null set N the restriction off to S - N has its values in a separable set , the

SES SES If for some null set N the restriction off to S - N has its values in a separable set , the

**function**f is said to be jl - essentially separably valued . Expressions such as essentially compact valued are**defined**similarly .Page 182

Q.E.D. Note : The unique u - integrable

Q.E.D. Note : The unique u - integrable

**function**of Theorems 2 and 7 is called the Radon - Nikodým derivative of 2 with respect to u and is often denoted by da / du . Thus da / du is**defined**u - almost everywhere by the formula ( ( da 2 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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