## Linear Operators: General theory |

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Page 89

Occasionally it is necessary to consider metric

complete . ... Then X is isomorphic and isometric with a dense linear subspace of

an F - space Æ . The space X is uniquely determined up to isometric

isomorphism .

Occasionally it is necessary to consider metric

**linear spaces**which are notcomplete . ... Then X is isomorphic and isometric with a dense linear subspace of

an F - space Æ . The space X is uniquely determined up to isometric

isomorphism .

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 I " . is the

scalars az , . . . , Olon with the norm lx | = sup laila 1Sisn 4 . The space l , is

defined for 1 Sp < oo as the

which the ...

The space I " . is the

**linear space**of all ordered n - tuples x = [ Q , . . . , an ] ofscalars az , . . . , Olon with the norm lx | = sup laila 1Sisn 4 . The space l , is

defined for 1 Sp < oo as the

**linear space**of all sequences x = { n } of scalars forwhich the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

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