## Linear Operators: General theory |

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Page 182

Thus da du is

Thus da du is

**defined**u - almost everywhere by the formula ( d . à ( E ) ( s ) ; u ( ds ) , Εε Σ . Idu We close this section with a generalization of many “ change of variable " theorems . E c Ez ; 8 LEMMA .Page 240

It is evident that if we

It is evident that if we

**define**the set function u on E by placing u ( E ) = 0 if E + and u ... The space B ( S ) is**defined**for an arbitrary set S and consists of all bounded scalar functions on S. The norm is given by lit = sup ...Page 241

**defined**on the o - field B of all Borel sets in S. The norm ( ul is the total variation v ( u , S ) . 18. The space L ( S , E , u ) is**defined**for any real number p , isp < oo , and any positive measure space ( S , E , u ) .### 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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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