## Linear Operators: General theory |

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Page 177

E C En may be

E C En may be

**assumed**that 2 is real valued . A real valued set function can be represented as the difference of its positive and negative variations ( 4.11 ) and so we may also**assume**that 2 is positive .Page 178

Thus we may and shall

Thus we may and shall

**assume**that v ( u , F ) < oo and v ( a , F ) < 0 . Since g is a - measurable there is a sequence { gn } of simple functions converging to g ( s ) for every s in F except on a set E C F with v ( 2 , E ) = 0 ( by ...Page 535

35

35

**Assume**u ( S ) 1. Put exp { / s log | f ( s ) \ n ( ds ) = 110 for f in L , ( S , E , u ) . Show that if f and g are non - negative functions in L ( S , E , u ) , then If + glo lilo + glo . ( Hint . Use Exercise III.9.29 . ) ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero