## Linear Operators: General theory |

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Page 484

The product of a

is

, BEQ , and let T , U be

The product of a

**weakly compact**linear operator and a bounded linear operatoris

**weakly compact**. PROOF . Let T , U € B ( X , Y ) , We B ( Y , 3 ) , Ve B ( 3 , X ) , &, BEQ , and let T , U be

**weakly compact**. The following inclusions then follow ...Page 494

2 we conclude that T * maps the unit sphere of X * into a conditionally

from C ( S ) to X , then T sends weak Cauchy sequences into strongly convergent

...

2 we conclude that T * maps the unit sphere of X * into a conditionally

**weakly****compact**set of rca ( S ) , and therefore T * is a ... If T is a**weakly compact**operatorfrom C ( S ) to X , then T sends weak Cauchy sequences into strongly convergent

...

Page 507

Let ( S , E , u ) be a o - finite positive measure space , and let T be a

there exists a u - essentially unique bounded measurable function a ( • ) on S to a

...

Let ( S , E , u ) be a o - finite positive measure space , and let T be a

**weakly****compact**operator on L ( S , E , u ) to a separable subset of the B - space X . Thenthere exists a u - essentially unique bounded measurable function a ( • ) on S to a

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero