## Linear Operators, Part 1 |

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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 sp Soo and p - 1 + 9-1 [ Q , . . . , an ] determined by the equation X * æ * x =

Σαβ :, X = { Bi } elani is an

clear that the mapping is an isomorphism . To see that it is also an

...

If1 sp Soo and p - 1 + 9-1 [ Q , . . . , an ] determined by the equation X * æ * x =

Σαβ :, X = { Bi } elani is an

**isometric**isomorphism of ( 1 % ) * onto 19 . Proof . It isclear that the mapping is an isomorphism . To see that it is also an

**isometric**map...

Page 313

The correspondence U : M1 → Mly is an

onto ca ( S1 , E2 ) . ( c ) If E , is in £ , then v ( u , E , ) = v ( U ( 411 ) , E ) for all ) My

in ba ( S1 , E ) . Proof . Recalling that t is an isomorphism of onto E , it is clear that

...

The correspondence U : M1 → Mly is an

**isometric**isomorphism of ba ( S ,, £ ; )onto ca ( S1 , E2 ) . ( c ) If E , is in £ , then v ( u , E , ) = v ( U ( 411 ) , E ) for all ) My

in ba ( S1 , E ) . Proof . Recalling that t is an isomorphism of onto E , it is clear that

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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

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