## Linear Operators, Part 1 |

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Page 269

in the product topology is equivalent to the convergence of the scalars {f,(s)} for

each s e S. The relative product ... f(s), fe F. The

topology generated by the neighborhoods N(fo; B, e) = {ffe C(S), wo(f-fo) < e, wo

e B} ...

in the product topology is equivalent to the convergence of the scalars {f,(s)} for

each s e S. The relative product ... f(s), fe F. The

**weak topology**of C(S) is thetopology generated by the neighborhoods N(fo; B, e) = {ffe C(S), wo(f-fo) < e, wo

e B} ...

Page 430

employed the concepts of weak sequential compactness (II.3.25) and

compactness in the 3:" (or

weak compactness ...

**Weak Topologies**. Weak Compactness We have already introduced andemployed the concepts of weak sequential compactness (II.3.25) and

compactness in the 3:" (or

**weak**)**topology**. There is at least one other type ofweak compactness ...

Page 477

strong, and the

operator

strong operator

uniform ...

strong, and the

**weak**operator**topologies**in B(t, y)). 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 ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero