## 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 483

If T is weakly

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 ) x ( TS ) . According to Theorem V.4.5 ...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 $ * of Y * is Y -**compact**( V.4.2 ) , it follows from Lemma 7 and ...### 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 Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero