## Linear Operators: General theory |

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Page 483

If T is

If T is

**weakly compact**, then TS is compact in the Y * topology of Y and thus z ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is**weakly compact**, ( i ) yields 7 ** ( S , ) Ç ( TS ) .Page 484

Linear combinations of

Linear combinations of

**weakly compact**operators are**weakly compact**. The product of a**weakly compact**linear operator and a bounded linear operator is**weakly compact**. PROOF . Let T , U € B ( X , Y ) , We B ( Y , 3 ) , V € B ( 3 , X ) ...Page 507

Let ( S , E , u ) be a o - finite positive measure space , and let T be 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. Then there exists a u - essentially unique bounded measurable function x ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero