## Linear Operators: General theory |

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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 ) Cx ( TS ) . According to Theorem V.4.5 , S , = S ** ...Page 494

From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally

S ) to X , then T sends weak Cauchy sequences into strongly convergent ...

From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally

**weakly compact**set of rca ( S ) , and ... If T is a**weakly compact**operator from 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 x ( s ) 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 ( s ) on S to a

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero