## Linear Operators: General theory |

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Page 425

If x is the natural embedding of a B - space X into X ** , then xx is X * -dense in ***

, PROOF . The X * -closure of X ( X ) is a subspace of X ** , which , by Theorem 5 ,

contains the unit

If x is the natural embedding of a B - space X into X ** , then xx is X * -dense in ***

, PROOF . The X * -closure of X ( X ) is a subspace of X ** , which , by Theorem 5 ,

contains the unit

**sphere**of X ** . It follows immediately that it contains every ...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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad 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 semi-group 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