## Linear Operators: General theory |

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Page 477

strong, and the weak

strong

uniform ...

strong, and the weak

**operator topologies**in 3)). It is evident that the uniform**operator topology**is stronger than the strong**operator topology**, and that thestrong

**operator topology**is stronger than the weak**operator topology**. With itsuniform ...

Page 511

is algebraically isomorphic to a subspacc of the product PMj where 3)x = 3),

under the mapping T P^j Tx. Show that the strong

?)) is identical with the usual product

in ...

is algebraically isomorphic to a subspacc of the product PMj where 3)x = 3),

under the mapping T P^j Tx. Show that the strong

**topology**of**operators**in jB(X,?)) is identical with the usual product

**topology**where the strong**topology**is takenin ...

Page 512

is compact in the strong

sequentially compact in the weak

the weak

3J) is ...

is compact in the strong

**operator topology**. If 3) is also separable, A issequentially compact in the weak

**operator topology**if and only if A is compact inthe weak

**operator topology**. 6 If 3) is reflexive, then the closed unit sphere of B($,,3J) is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero