## 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 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 ) . According to Theorem V.4.5 , S , = S ** ...Page 484

The product of a

is

BED , and let T , U be

...

The product of a

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

**weakly compact**. PROOF . Let T , U e B ( X , Y ) , We B ( Y , 3 ) , V € B ( 3 , X ) , ,BED , and let T , U be

**weakly compact**. The following inclusions then follow from...

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 x ( • ) 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 x ( • ) on S to a

...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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