## Linear Operators: General theory |

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Page 72

I.11 ) is a

I.11 ) is a

**B**-**space**( or an F - space ) with the metric \ x + 31 = inf ( x + xl . 263 ( Hint : Given a Cauchy sequence in X / 3 , define a subsequence for which Xx — Xx + 2 + 31 < 2 , k = 1 , 2 , ... , and show that a Cauchy \ 1 - k ...Page 89

In the definitions of F- and

In the definitions of F- and

**B**-**spaces**, we required the spaces to be complete in their metric topology . Occasionally it is necessary to consider ... Let X be a linear space satisfying properties ( i ) and ( ii ) of Definition 1.10 .Page 398

168 ] showed that every X - convergent sequence in X * is actually X ** - convergent , and that if y is a separable

168 ] showed that every X - convergent sequence in X * is actually X ** - convergent , and that if y is a separable

**B**-**space**, then any operator in B ( X , Y ) is weakly compact . Goodner [ 1 ; p . 103 ] proved that the unit sphere in ...### 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