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

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

Page 484

The product of a

is

, ß€ 0 , 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 ) , a, ß€ 0 , 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

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional 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 neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect 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