## Linear Operators: General theory |

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Page 17

A topological space X is said to be locally

neighborhood whose closure is

intersection property if every finite subfamily has a non - void intersection . A

subset of X is called ...

A topological space X is said to be locally

**compact**if every point has aneighborhood whose closure is

**compact**. A family of sets has the finiteintersection property if every finite subfamily has a non - void intersection . A

subset of X is called ...

Page 483

If T is weakly

) 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 , = $ ** ...

Page 494

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

weakly

By Theorem 4.8 this implies that T is a weakly

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

weakly

**compact**set of rca ( S ) , and therefore T * is a weakly**compact**operator .By Theorem 4.8 this implies that T is a weakly

**compact**operator . Q.E.D. given ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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 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