## Linear Operators, Part 1 |

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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 Ã . The space Ž 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 Ã . The space Ž is uniquely determined up to isometric isomorphism

.

Page 91

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

, a normed

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

have ...

Thus every complete linear metric space can be metrized to be 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**. Wehave ...

Page 239

The space I " is the

, ... , Own with the norm = suplovila 1Sisn 4 . The space l , is defined for 1 sp < oo

as the

The space I " is the

**linear space**of all ordered n - tuples [ Q , ... , an ] of scalars dq, ... , Own with the norm = suplovila 1Sisn 4 . The space l , is defined for 1 sp < oo

as the

**linear space**of all sequences x = { n } of scalars for which the norm n = 1 ...### 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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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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 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 transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero