## 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

following ...

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 . In such cases , thefollowing ...

Page 91

Thus every complete linear metric space can be metrized to be an F - space .

Further , a normed

equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in

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 someequivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in

**linear spaces**.Page 239

The space 1 " is the

Qy , . . . , Chen with the norm = sup loilo 1Sisn 4. The space l , is defined for 1 < p

< oo as the

The space 1 " is the

**linear space**of all ordered n - tuples ( Q2 , ... , obn ] of scalarsQy , . . . , Chen with the norm = sup loilo 1Sisn 4. The space l , is defined for 1 < p

< oo as the

**linear space**of all sequences x = { en } of scalars for which the ...### 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