## 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 1 ' " is the

The space 1 ' " is the

**linear space**of all ordered n - tuples [ 01 , an ] of scalars ( 2 , ... , Chon with the norm | x ) = sup loilo 1Sisn 4. The space lp is defined for 1 sp < oo as the**linear space**of all sequences x { an } of ...### 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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### Common terms and phrases

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