## Linear Operators: General theory |

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Page 89

Completion of spaces . 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 metric

Completion of spaces . 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 metric

**linear spaces**which are not complete .Page 91

Thus every complete linear metric space can be metrized to be an F - space . Further , a normed

Thus every complete linear metric space can be metrized to be an F - space . Further , a normed

**linear space**is a B - space provided it is complete under some equivalent metric . See also van Dantzig ( 1 ] , [ 2 ] .Page 239

The space I ' " is the

The space I ' " is the

**linear space**of all ordered n - tuples [ , . an ] of scalars Oj , . ... , an with the norm = sup lail . 1 Sisn 4 . The space 1 , is defined for 1 p < oo as the**linear space**of all sequences x = { an } of scalars ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

91 other sections not shown

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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero