## Linear Operators: General theory |

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Results 1-3 of 64

Page 12

Nelson Dunford, Jacob T. Schwartz. listed in Lemma 5 , so that the family of

complements of elements of F is a topology . The set A is the

topology . PROOF . Statements 10 ( b ) , ( d ) show that $ , X€ F , and 10 ( a )

shows ...

Nelson Dunford, Jacob T. Schwartz. listed in Lemma 5 , so that the family of

complements of elements of F is a topology . The set A is the

**closure**of A in thistopology . PROOF . Statements 10 ( b ) , ( d ) show that $ , X€ F , and 10 ( a )

shows ...

Page 432

To see this , let m be an arbitrary integer ; since w * * is in the X * -

) , there is an element zm € A such that ( = m ) = a * ( * ) < 1m , į = 1 , . . . , n . Since

A is weakly sequentially compact , a subsequence of { zm } will converge ...

To see this , let m be an arbitrary integer ; since w * * is in the X * -

**closure**of x ( A) , there is an element zm € A such that ( = m ) = a * ( * ) < 1m , į = 1 , . . . , n . Since

A is weakly sequentially compact , a subsequence of { zm } will converge ...

Page 511

2 A set AC B ( X , Y ) is compact in the weak operator topology if and only if it is

closed in the weak operator topology and the weak

compact for each x € X . A set AC B ( X , Y ) is compact in the strong operator

topology if ...

2 A set AC B ( X , Y ) is compact in the weak operator topology if and only if it is

closed in the weak operator topology and the weak

**closure**of Ax is weaklycompact for each x € X . A set AC B ( X , Y ) is compact in the strong operator

topology if ...

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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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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