## Linear Operators: General theory |

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Results 1-3 of 81

Page 511

Show that the strong topology of operators in B ( X , Y ) is identical with the usual

product topology where the strong topology is taken in each Yz , and that the

.

Show that the strong topology of operators in B ( X , Y ) is identical with the usual

product topology where the strong topology is taken in each Yz , and that the

**weak**operator topology is that where each factor is taken to have the Y * topology.

Page 512

If Y is also separable , A is sequentially compact in the

and only if A is compact in the

the closed unit sphere of B ( X , Y ) is compact in the

If Y is also separable , A is sequentially compact in the

**weak**operator topology ifand only if A is compact in the

**weak**operator topology . 6 If Y is reflexive , thenthe closed unit sphere of B ( X , Y ) is compact in the

**weak**operator topology .Page 858

9 ( 7 ) Wiener measure space , ( 405 ,

67 ) properties , II . 3 . 26 – 27 ( 68 ) in special spaces , IV . 15

additivity , definition , ( 318 ) and strong , IV . 10 . 1 ( 318 )

II .

9 ( 7 ) Wiener measure space , ( 405 ,

**Weak**convergence , definition , II . 3 . 25 (67 ) properties , II . 3 . 26 – 27 ( 68 ) in special spaces , IV . 15

**Weak**countableadditivity , definition , ( 318 ) and strong , IV . 10 . 1 ( 318 )

**Weak**limit , definition ,II .

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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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### Common terms and phrases

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