## Linear Operators: General theory |

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Page 151

**Consequently**, sup || n Imlo Slim sup ( SF , \ n ( 8 ) –Im ( 8 ) pulp , ds ) } " + 2ello Slim sup [ { SE . \\ . ( 8 ) –f ( s ) pulu , ds ) " " " + { SE . \ tm ( 8 ) – † ( 8 ) | ” 0 ( ja , ds ) " ' ] + 2ello 2ɛllo , so that lim in - Iml ...Page 451

Since the function 1 g ( a , x , y ) { t ( x + ay ) -2 ( x ) + ( x - ay ) } a 2n , i , i { is ( by Lemma 1 ) the sum of two monotone increasing functions of a it has this same property for all x , y € X.

Since the function 1 g ( a , x , y ) { t ( x + ay ) -2 ( x ) + ( x - ay ) } a 2n , i , i { is ( by Lemma 1 ) the sum of two monotone increasing functions of a it has this same property for all x , y € X.

**Consequently**, letting = x \ x ...Page 627

**Consequently**, by the Weierstrass approximation theorem , and Corollary III.10.6 , So hlu ) g ( u ) du = S'h ( u ) G ( du ) , he C ( 0 , 1 ) , where G ( E ) = Seg ( u ) du . Since Lebesgue measure is regular so is G and thus , by the ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero