## Linear Operators: General theory |

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Page 426

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

closed unit

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

define 1 | ( x ...

Nelson Dunford. O 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 { xn } be a countable dense subset of X , and

define 1 | ( x ...

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

Lemma 7 and Lemma 1.5.7 that T * S * is ... If S , S ** are the closed unit

in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by ...

Since the closed unit

**sphere**S * of Y * is Y - compact ( V.4.2 ) , it follows fromLemma 7 and Lemma 1.5.7 that T * S * is ... If S , S ** are the closed unit

**spheres**in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero