## Linear Operators, Part 1 |

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12

and ( T , ET , 2 ) . Let E be a o - null set in R . Then for 2 - almost all t , the set E ( t

) = { s [ s , t ] € E } is a u - null set . Proof . By

12

**LEMMA**. Let ( R , ER , 0 ) be the product of finite measure spaces ( S , E , u )and ( T , ET , 2 ) . Let E be a o - null set in R . Then for 2 - almost all t , the set E ( t

) = { s [ s , t ] € E } is a u - null set . Proof . By

**Lemma**11 it may be assumed that ...Page 586

By

chosen such that T ( 0 ) - T ( u ) ... theory of analytic functions ( cf . III . 14 ) that R (

2 ; T ( u ) ) is an analytic function of u , lul < o , for each à € U . Q . E . D . 5

...

By

**Lemma**3 , there is a & , such that if sul < $ , then U CO ( T ( u ) ) . Let S < d bechosen such that T ( 0 ) - T ( u ) ... theory of analytic functions ( cf . III . 14 ) that R (

2 ; T ( u ) ) is an analytic function of u , lul < o , for each à € U . Q . E . D . 5

**LEMMA**...

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Thus , to prove the

the

Thus , to prove the

**lemma**, it will suffice to show that it is true if k is even . If k = 1the

**lemma**has already been proved (**Lemma**6 ) . We shall thus suppose that the**lemma**has been proved for the integer n and shall conclude its validity for the ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero