## Linear Operators, Part 1 |

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Page 483

If T is

) is compact and hence closed in the Y * topology of Y * * . Thus if T is

...

If T is

**weakly compact**, then TS is compact in the Y * topology of Y and thus x ( TS) is compact and hence closed in the Y * topology of Y * * . Thus if T is

**weakly****compact**, ( i ) yields T * * ( S ) C x ( TS ) . According to Theorem V . 4 . 5 , S = S * *...

Page 494

2 we conclude that 7 * 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 7 * 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

may be noted that the next theorem applies to every continuous linear map of Lj (

S , E , u ) into a separable reflexive space . 10 THEOREM . Let ( S , E , u ) be a o -

finite positive measure space , and let T be a

may be noted that the next theorem applies to every continuous linear map of Lj (

S , E , u ) into a separable reflexive space . 10 THEOREM . Let ( S , E , u ) be a o -

finite positive measure space , and let T be a

**weakly compact**operator on L ( S ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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