## Linear Operators: General theory |

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Page 269

The relative product topology on F is the topology generated by the

neighborhoods Ar(/0; A, e) = {/|/ € F, |/(*)— /0(*)| < s, ... e F. The

C(S) is the topology generated by the neighborhoods N(f0; B, e) = {j\f e C{S), \x*(f-

f0)\ < e, x* e ...

The relative product topology on F is the topology generated by the

neighborhoods Ar(/0; A, e) = {/|/ € F, |/(*)— /0(*)| < s, ... e F. The

**weak topology**ofC(S) is the topology generated by the neighborhoods N(f0; B, e) = {j\f e C{S), \x*(f-

f0)\ < e, x* e ...

Page 477

strong, and the

operator

strong operator

uniform ...

strong, and the

**weak**operator**topologies**in 3)). It is evident that the uniformoperator

**topology**is stronger than the strong operator**topology**, and that thestrong operator

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

Page 512

is compact in the strong operator

sequentially compact in the

the

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

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### 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