## 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 658

Thus , the mathematician is led to the problem of determining whether or not the

various of its generalizations which arise in the theory of stochastic processes .

Thus , the mathematician is led to the problem of determining whether or not the

**limit**( iii ) exists . The next four sections are devoted to a study of this problem andvarious of its generalizations which arise in the theory of stochastic processes .

Page 683

from Lemma 10 and Corollary 5.3 that the

Corollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

from Lemma 10 and Corollary 5.3 that the

**limit**m = lim Mn exists in ca ( , u ) . ByCorollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

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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