## Linear Operators: General theory |

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Page 182

Thus da du is

Εε Σ . We close this section with a generalization of many change of variable "

theorems . Εε Σ .; 8 LEMMA . Let S , and S , be sets and $ a mapping of S , into S ,

.

Thus da du is

**defined**u - almost everywhere by the formula lai 2 ( E ) ( s ) u ( ds ) ,Εε Σ . We close this section with a generalization of many change of variable "

theorems . Εε Σ .; 8 LEMMA . Let S , and S , be sets and $ a mapping of S , into S ,

.

Page 240

It is evident that if we

uld ... The space B ( S ) is

bounded scalar functions on S. The norm is given by Wit = sup \ | ( s ) / . SES

It is evident that if we

**define**the set function u on by placing u ( E ) = c if E +6 anduld ... The space B ( S ) is

**defined**for an arbitrary set S and consists of allbounded scalar functions on S. The norm is given by Wit = sup \ | ( s ) / . SES

**defined**on ...Page 534

function f

Sot xn f ( x ) dx , n 20 , is a bounded map of L , into l , such that TT * is the map {

an } + { by }

that ...

function f

**defined**on the interval ( 0 , 1 ) into the sequence { an }**defined**by anSot xn f ( x ) dx , n 20 , is a bounded map of L , into l , such that TT * is the map {

an } + { by }

**defined**by а ; bn = { s = on + i + 1 Show that T has norm Vă . Showthat ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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 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