## Linear Operators: General theory |

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Page 430

employed the concepts of

compactness in the X * ( or

**Weak**Topologies .**Weak**Compactness We have already introduced andemployed the concepts of

**weak**sequential compactness ( II . 3 . 25 ) andcompactness in the X * ( or

**weak**) topology . There is at least one other type of**weak**...Page 477

strong , and the

uniform operator topology is stronger than the strong operator topology , and that

the strong operator topology is stronger than the

...

strong , and the

**weak**operator topologies in B ( X , Y ) . It is evident that theuniform operator topology is stronger than the strong operator topology , and that

the strong operator topology is stronger than the

**weak**operator topology . With its...

Page 511

2 A set AC B ( X , Y ) is compact in the

closed in the

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 isclosed 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

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group 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