## Linear Operators: General theory |

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Page

Topologies . Compactness and Reflexivity . . . . . . 423 5 .

Metrizability . Unbounded Sets . . . . . 425 6 .

Compactness . . . 7 .

**Weak**Topologies . Definitions and Fundamental Properties . 418 4 .**Weak**Topologies . Compactness and Reflexivity . . . . . . 423 5 .

**Weak**Topologies .Metrizability . Unbounded Sets . . . . . 425 6 .

**Weak**Topologies .**Weak**Compactness . . . 7 .

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 e 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 e 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 | |

80 other sections not shown

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### Common terms and phrases

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 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