## Linear Operators, Part 1 |

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Page 182

Thus dh / du is

3 ) Mcds ) . We close this section with a generalization of many change of

variable " theorems . 8 LEMMA . Let S , and S , be sets and $ a mapping of S ,

into S2 .

Thus dh / du is

**defined**u - almost everywhere by the formula ( da 2 ( E ) = Εε Σ . (3 ) Mcds ) . We close this section with a generalization of many change of

variable " theorems . 8 LEMMA . Let S , and S , be sets and $ a mapping of S ,

into S2 .

Page 240

It is evident that if we

u ( $ ) 0 , then a bounded function is ... The space B ( S ) is

set S and consists of all bounded scalar functions on S. The norm is given by ...

It is evident that if we

**define**the set function u on by placing u ( E ) = 0 if E + $ andu ( $ ) 0 , then a bounded function is ... The space B ( S ) is

**defined**for an arbitraryset S and consists of all bounded scalar functions on S. The norm is given by ...

Page 534

function f

" | ( x ) dx , n 20 , is a bounded map of L , into l , such that TT * is the map { an } > {

bon }

function f

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

bon }

**defined**by а ; ba Σ S ( x - j = on + + 1 Show that T has norm Vr . Show ...### 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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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