## Linear Operators: General theory |

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Page 212

Let M be a finite

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

a family F of closed sets if each Fe F has

Let M be a finite

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

a family F of closed sets if each Fe F has

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

T : L , → L , is

Similarly T , 2 T , or T , ST , means that T1 - T , 20 . 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 : L , → L , is

**positive**and write T 20 when | eL , and I 20 imply that Tf 20.Similarly T , 2 T , or T , ST , means that T1 - T , 20 . 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

20 , j = 1 , 2 , and let 11 X - 181 , 12 = 21–1824 be decompositions of fı and into

...

We now show that T , is additive on

**positive**elements . Let t , € L , ( S , E , u ) , t ,20 , j = 1 , 2 , and let 11 X - 181 , 12 = 21–1824 be decompositions of fı and into

**positive**functions . Then Efli + E821 is a decomposition of f1 +12 , and so To ( 1 +...

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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