## Linear Operators, Part 1 |

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Page 146

. Q.E.D. 5 COROLLARY . If ( S , E , u ) is a measure space , the spaces TM ( S , E ,

u , X ) and M ( S , E , u , X ) of totally measurable and measurable functions are ...

**Consequently**, \ in - 1 = \ In - In , 1+ \ in , -1 < for n N , so that In → f in u - measure. Q.E.D. 5 COROLLARY . If ( S , E , u ) is a measure space , the spaces TM ( S , E ,

u , X ) and M ( S , E , u , X ) of totally measurable and measurable functions are ...

Page 151

Nelson Dunford, Jacob T. Schwartz. Selt ( s ) P 0 ( , ds ) < oo and lim Salto ( 8 ) —

t ( 3 ) pd ( u , ds ) = 0 follow easily from Corollary 13 ( b ) and Theorem 3.6 .

1 / 0 ...

Nelson Dunford, Jacob T. Schwartz. Selt ( s ) P 0 ( , ds ) < oo and lim Salto ( 8 ) —

t ( 3 ) pd ( u , ds ) = 0 follow easily from Corollary 13 ( b ) and Theorem 3.6 .

**Consequently**, lim sup \ fn - Imlo Slim sup { St. \\ , ( s ) —Im ( s ) pulp , ds ) " ' + 2 ¢1 / 0 ...

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0 ; so that , if w > 0 , is chosen large enough , we will have S® ee - wt y ( t ) dt = p

< 1 .

**Consequently**, ( i ) , ... , ( v ) are proved inductively for all n . We now ...**Consequently**, by III.6.16 , lim Soehtvy ( t ) dt = lim Se - 1 - w ) t { e - art y ( t ) } dt =0 ; so that , if w > 0 , is chosen large enough , we will have S® ee - wt y ( t ) dt = p

< 1 .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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

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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 isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space 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