## 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 Ž . 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 Ž . The space ž 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 dy , . . . , Chon with the norm lx | = sup laila 1 Sisn 4 . The space lo is

defined for 1 p < oo as the

which ...

The space I " . is the

**linear space**of all ordered n - tuples x = [ 0 , . . . , Cen ] ofscalars dy , . . . , Chon with the norm lx | = sup laila 1 Sisn 4 . The space lo is

defined for 1 p < oo as the

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect 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