## Linear Operators: General theory |

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Page 68

The space X is said to be weakly complete if every weak Cauchy sequence has a

weak

such a way that they become Hausdorff spaces in which the notion of

convergence ...

The space X is said to be weakly complete if every weak Cauchy sequence has a

weak

**limit**. In Chapter V , a topology is introduced in certain linear spaces insuch a way that they become Hausdorff spaces in which the notion of

convergence ...

Page 126

We define the

inf E , un E. lim sup En nu Em 72 n = 1 men n n = 1 man If lim inf , En = lim sup ,

En , { En } is said to be convergent , and we write the common value of the

We define the

**limit**inferior and the**limit**superior of { En } by the equations or liminf E , un E. lim sup En nu Em 72 n = 1 men n n = 1 man If lim inf , En = lim sup ,

En , { En } is said to be convergent , and we write the common value of the

**limit**...Page 127

It is evident that such a sequence has the

sequence { En } is one for which En ? En + 1 , n = 1 , 2 , .... Such a sequence has

the

non ...

It is evident that such a sequence has the

**limit**U21 En . A non - increasingsequence { En } is one for which En ? En + 1 , n = 1 , 2 , .... Such a sequence has

the

**limit**n = 1 En . A monotone sequence is which is either non - decreasing ornon ...

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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