## Linear Operators: General theory |

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A topological space X is said to be locally

A topological space X is said to be locally

**compact**if every point has a neighborhood whose closure is**compact**. A family of sets has the finite intersection property if every finite subfamily has a non - void intersection .Page 485

( Gantmacher ) An operator in B ( X , Y ) is weakly

( Gantmacher ) An operator in B ( X , Y ) is weakly

**compact**if and only if its adjoint is weakly**compact**. PROOF . Let T be weakly**compact**. Since the closed unit sphere S * of Y * is Y -**compact**( V.4.2 ) , it follows from Lemma 7 and ...Page 494

From IV.10.2 we conclude that T * maps the unit sphere of X * into a conditionally 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 .### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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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