## Linear Operators: General theory |

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Page 426

Nelson Dunford. 1 THEOREM . If X is a B - space , then the X topology of the

closed unit

PROOF . If X is separable , let { wn } be a countable dense subset of X , and

define 1 ...

Nelson Dunford. 1 THEOREM . If X is a B - space , then the X topology of the

closed unit

**sphere**S * of X * is a metric topology if and only if X is separable .PROOF . If X is separable , let { wn } be a countable dense subset of X , and

define 1 ...

Page 458

5 If the closed unit

finite number of extremal points , then X is not isometrically isomorphic to the

conjugate of any B - space . 6 Let S be a topological space , and let C ( S ) be the

B ...

5 If the closed unit

**sphere**of an infinite dimensional B - space X contains only afinite number of extremal points , then X is not isometrically isomorphic to the

conjugate of any B - space . 6 Let S be a topological space , and let C ( S ) be the

B ...

Page 485

Since the closed unit

If S , S * * are the closed unit

natural embedding of X into X * * , then by Theorem V . 4 . 5 , xS is X * - dense in

S ...

Since the closed unit

**sphere**S * of Y * is Y - compact ( V . 4 . 2 ) , it follows from ...If S , S * * are the closed unit

**spheres**in X , X * * , respectively , and if x is thenatural embedding of X into X * * , then by Theorem V . 4 . 5 , xS is X * - dense in

S ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 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 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