## Linear Operators: General theory |

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Page 424

( Alaoglu ) The closed

compact in the X topology of X * . Proof . By Definition II . 3 . 5 , the

X * is the set J j €X * , j ( r ) Se , and thus the theorem follows from Lemma 1 .

( Alaoglu ) The closed

**unit sphere**in the conjugate space X * of the B - space X iscompact in the X topology of X * . Proof . By Definition II . 3 . 5 , the

**unit sphere**inX * is the set J j €X * , j ( r ) Se , and thus the theorem follows from Lemma 1 .

Page 425

A B - space is reflexive if and only if its closed

topology . PROOF . Let X be a reflexive B - space , and let ~ be the natural

embedding of X onto X * * . Then x and x - 1 are isometries , and x maps the

closed ...

A B - space is reflexive if and only if its closed

**unit sphere**is compact in the weaktopology . PROOF . Let X be a reflexive B - space , and let ~ be the natural

embedding of X onto X * * . Then x and x - 1 are isometries , and x maps the

closed ...

Page 512

6 If Y is reflexive , then the closed

operator topology . Conversely , if the closed

in the weak operator topology , Y is reflexive . 7 Define the BWO topology for B ...

6 If Y is reflexive , then the closed

**unit sphere**of B ( X , Y ) is compact in the weakoperator topology . Conversely , if the closed

**unit sphere**of B ( x , y ) is compactin the weak operator topology , Y is reflexive . 7 Define the BWO topology for B ...

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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