## Linear Operators: General theory |

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Page 186

Thus , the field of Lemma 1 is a o - field and the measure U of Lemma 1 is

countably additive on Ø . Consequently , the restriction of u to the o ... Let ( S , E ,

u ) be the product of finite

uz ) .

Thus , the field of Lemma 1 is a o - field and the measure U of Lemma 1 is

countably additive on Ø . Consequently , the restriction of u to the o ... Let ( S , E ,

u ) be the product of finite

**positive measure spaces**( S1 , E1,47 ) and ( S2 , E2 ,uz ) .

Page 302

This clearly is a partial ordering ( I.2.1 ) in Lp . We now establish completeness (

1.12 ) with respect to this ordering . 22 THEOREM . Let ( S , E , u ) be a

...

This clearly is a partial ordering ( I.2.1 ) in Lp . We now establish completeness (

1.12 ) with respect to this ordering . 22 THEOREM . Let ( S , E , u ) be a

**positive****measure space**. Then the real partially ordered space L ( S , E , j ) , 1 p < oo , is a...

Page 725

1 n - 1 lim sup I u ( p = re ) S Ku ( e ) n + 00 nj = 0 for each set e of finite u -

measure . ( Hint . Consider the map f ( s ) → XA ( s ) / ( ps ) for each A € E with u (

A ) < 00 . ) 37 Let ( S , E , u ) be a

...

1 n - 1 lim sup I u ( p = re ) S Ku ( e ) n + 00 nj = 0 for each set e of finite u -

measure . ( Hint . Consider the map f ( s ) → XA ( s ) / ( ps ) for each A € E with u (

A ) < 00 . ) 37 Let ( S , E , u ) be a

**positive measure space**, and T a non - negative...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 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