## Linear Operators: General theory |

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Page 262

Since the integral Ss / ( s ) u ( ds )

|| ( s ) 0 ( 4 , 5 ) , 8 ISs g ( s ) u ( ds ) – ... there is an isometric isomorphism

between C * ( S ) and rba ( S ) such that corresponding elements q * and u

the ...

Since the integral Ss / ( s ) u ( ds )

**satisfies**the inequality 115 11 3 ) u ( ds ) < sup|| ( s ) 0 ( 4 , 5 ) , 8 ISs g ( s ) u ( ds ) – ... there is an isometric isomorphism

between C * ( S ) and rba ( S ) such that corresponding elements q * and u

**satisfy**the ...

Page 495

Evidently B ( h ) C B. , and it is clear that B ( h )

above so that B ( h ) = B .. This proves that if h e C ( S ) and f e Bo , then sh e Bo .

Now let f be a fixed element of Bo , and let B ( 1 ) = { g € B lg € B. } . We have just

...

Evidently B ( h ) C B. , and it is clear that B ( h )

**satisfies**properties ( i ) and ( ii )above so that B ( h ) = B .. This proves that if h e C ( S ) and f e Bo , then sh e Bo .

Now let f be a fixed element of Bo , and let B ( 1 ) = { g € B lg € B. } . We have just

...

Page 613

... the most general continuous real ( or complex ) function of the nonnegative

real variable t which

operator valued functions defined on the range t 2 0 , which

( i ) T ( t ...

... the most general continuous real ( or complex ) function of the nonnegative

real variable t which

**satisfies**the functional ... the most general continuousoperator valued functions defined on the range t 2 0 , which

**satisfy**the equations( i ) T ( t ...

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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