## Linear Operators: General theory |

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Page 138

inn i = n

proving that u ( E ) = u ( En ) . Q . E . D . n = 1 14 THEOREM . Let u be a bounded

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**Consequently**, Lil v ( u , E ; ) < oo , and vlu , UE : ) = v ( u , E ; ) + 0 , as n → 00 .inn i = n

**Consequently**, - 1 Tu ( E ) – Eu ( E ; ) ] = lu ( UE ; ) = v ( u , u E ; ) → 0 ,proving that u ( E ) = u ( En ) . Q . E . D . n = 1 14 THEOREM . Let u be a bounded

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Page 151

2e1 / o Slim sup [ { S£1tu ( 9 ) — + 1 ( s ) | po ( 41 , ds ) + { SF , \ em ( s ) – + ( 8 ) |

po ( ja , ds ) } " > ] + Zelo = 281 / P so that lim In - imp = 0 . Since L ( S , E , u , X ) ...

**Consequently**, lim sup | | n – Imlı Slim sup { St . 144 ( 8 ) — tm ( s ) polu , ds ) " " +2e1 / o Slim sup [ { S£1tu ( 9 ) — + 1 ( s ) | po ( 41 , ds ) + { SF , \ em ( s ) – + ( 8 ) |

po ( ja , ds ) } " > ] + Zelo = 281 / P so that lim In - imp = 0 . Since L ( S , E , u , X ) ...

Page 369

most n + 1 points - 1 < h Sta . . . < tk Şl , and there are constants C , . . . , Ct with X

- \ cil = \ H , and in terms of which we may write an “ interpolation formula ” . f ( x )

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**Consequently**, unless xo ( t ) is a constant of absolute value 1 , C is a set of atmost n + 1 points - 1 < h Sta . . . < tk Şl , and there are constants C , . . . , Ct with X

- \ cil = \ H , and in terms of which we may write an “ interpolation formula ” . f ( x )

...

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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