## Linear Operators: General theory |

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Page 212

Let u be a finite

compact metric space S . A set A CS is said to be covered in the sense of Vitali by

a family F of closed sets if each FeF has

Let u be a finite

**positive**measure defined on the o - field of Borel sets of acompact metric space S . A set A CS is said to be covered in the sense of Vitali by

a family F of closed sets if each FeF has

**positive**u - measure and there is a**positive**...Page 304

T : Lp → L , is

Similarly T , 2T , or T , ST , means that Ti - T , 2 0 . It is readily seen that if T 20 ,

then T maps real functions into real functions . 25 LEMMA . Let ( S , E , u ) be a

T : Lp → L , is

**positive**and write T 2 0 when | eL , and f 2 0 imply that Tf 20 .Similarly T , 2T , or T , ST , means that Ti - T , 2 0 . It is readily seen that if T 20 ,

then T maps real functions into real functions . 25 LEMMA . Let ( S , E , u ) be a

**positive**...Page 305

We now show that T , is additive on

, j = 1 , 2 , and let fi = n - 181is ta = { 1 – 1821 be decompositions of f1 and 12 into

We now show that T , is additive on

**positive**elements . Let t ; e L ( S , E , u ) , t ; 20, j = 1 , 2 , and let fi = n - 181is ta = { 1 – 1821 be decompositions of f1 and 12 into

**positive**functions . Then { & i + Szi is a decomposition of f1 + 12 , and so To ( fi ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero