## Linear Operators: General theory |

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Page 182

Thus da du is

) , Εε Σ . Idu We close this section with a generalization of many “ change of

variable " theorems . E c Ez ; 8 LEMMA . Let S , and S , be sets and $ a mapping

of S ...

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 . Let S , and S , be sets and $ a mapping

of S ...

Page 240

It is evident that if we

u ( ° ) = 0 , then a bounded function ... The space B ( S ) is

set S and consists of all bounded scalar functions on S. The norm is given by lit ...

It is evident that if we

**define**the set function u on E by placing u ( E ) = 0 if E + andu ( ° ) = 0 , then a bounded function ... The space B ( S ) is

**defined**for an arbitraryset S and consists of all bounded scalar functions on S. The norm is given by lit ...

Page 241

( u , S ) . 18. The space L ( S , E , u ) is

and any positive measure space ( S , E , u ) . It consists of those u - measurable ...

**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 ) . It consists of those u - measurable ...

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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