## Linear Operators: General theory |

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Page 182

Thus dh du is

£ . ... an additive set function My on £ , the equation wy ( 0 - 1 ( E ) ) = Mg ( E )

additive ...

Thus dh du is

**defined**u - almost everywhere by the formula ( da , ( s ) u ( ds ) , E€£ . ... an additive set function My on £ , the equation wy ( 0 - 1 ( E ) ) = Mg ( E )

**defines**an additive set function My on Ex . Moreover ( a ) if Mz is countablyadditive ...

Page 240

It is evident that if we

uld ) = 0 , then a bounded ... The space B ( S ) is

consists of all bounded scalar functions on S. The norm is given by Wit = sup ...

It is evident that if we

**define**the set function u on by placing u ( E ) = c if E +6 anduld ) = 0 , then a bounded ... The space B ( S ) is

**defined**for an arbitrary set S andconsists of all bounded scalar functions on S. The norm is given by Wit = sup ...

Page 534

function f

an = L * a " } ( x ) dx , is a bounded map of L2 into ly such that TT * is the map { an

} + { bn }

function f

**defined**on the interval ( 0 , 1 ) into the sequence { an }**defined**by n 20 ,an = L * a " } ( x ) dx , is a bounded map of L2 into ly such that TT * is the map { an

} + { bn }

**defined**by b . = jaon + i + 1 Show that T has norm va . Show that the ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero