## Linear Operators: General theory |

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Page 88

The fact that the natural isomorphism of a B - space X into its second conjugate is

notion of the conjugate space . He used the term regular to describe what we ...

The fact that the natural isomorphism of a B - space X into its second conjugate is

**isometric**was proved by Hahn [ 3 ; p . 219 ] who was the first to formulate thenotion of the conjugate space . He used the term regular to describe what we ...

Page 247

If1 < p Soo and p - 1 + q - 1 = 1 , then the mapping [ 02 , . . . , On ] determined by

the equation x * i = 1 x * x = Edißi x = { B } } € 1 " , is an

l ) * onto 1 . PROOF . It is clear that the mapping is an isomorphism . To see that it

...

If1 < p Soo and p - 1 + q - 1 = 1 , then the mapping [ 02 , . . . , On ] determined by

the equation x * i = 1 x * x = Edißi x = { B } } € 1 " , is an

**isometric**isomorphism of (l ) * onto 1 . PROOF . It is clear that the mapping is an isomorphism . To see that it

...

Page 313

The correspondence U : My → Ma is an

onto ca ( S1 , E2 ) . ( c ) If E , is in & , then v ( un , Eq ) = v ( U ( 47 ) , E ) for all My

in ba ( S , ; ) . Proof . Recalling that t is an isomorphism of onto & , it is clear that

the ...

The correspondence U : My → Ma is an

**isometric**isomorphism of ba ( S1 , Ey )onto ca ( S1 , E2 ) . ( c ) If E , is in & , then v ( un , Eq ) = v ( U ( 47 ) , E ) for all My

in ba ( S , ; ) . Proof . Recalling that t is an isomorphism of onto & , it is clear that

the ...

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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