## Linear Operators: General theory |

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Page 147

If f is u - integrable then set ( s ) u ( ds ) = 0 for every E in E if and only if 1

If f is u - integrable then set ( s ) u ( ds ) = 0 for every E in E if and only if 1

**vanishes**almost everywhere . Proof . It is clear that a function**vanishing**almost everywhere is a ...Page 302

It is evidently sufficient to show that if { / s } is a set of functions in L , such that 0 St , & , for some goeL , then supa { f } exists in Lj . Further , since go is integrable it

It is evidently sufficient to show that if { / s } is a set of functions in L , such that 0 St , & , for some goeL , then supa { f } exists in Lj . Further , since go is integrable it

**vanishes**u - almost everywhere outside of ...Page 334

... This proves that W is continuous at every point of X. By ( ii ) , W

... This proves that W is continuous at every point of X. By ( ii ) , W

**vanishes**on a set which is dense in X ; consequently W**vanishes**identically . Q.E.D. We shall now consider some order properties of the space M ( 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