## Linear Operators: General theory |

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Page 269

Since every point in S gives rise to a continuous linear functional on C(S), it is

clear that the product topology of C(S) is weaker than the

The next theorem indicates a close relation between weak sequential ...

Since every point in S gives rise to a continuous linear functional on C(S), it is

clear that the product topology of C(S) is weaker than the

**weak topology**of C(S ).The next theorem indicates a close relation between weak sequential ...

Page 430

employed the concepts of weak sequential compactness (II. 3. 25) and

compactness in the X* (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 X* (or

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

Page 512

is also separable, A is sequentially compact in the

only if A is compact in the

closed unit sphere of "2) ) is compact in the

is also separable, A is sequentially compact in the

**weak**operator**topology**if andonly if A is compact in the

**weak**operator**topology**. 6 If ?) is reflexive, then theclosed unit sphere of "2) ) is compact in the

**weak**operator**topology**. Conversely ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite 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 continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space 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 valued function vector space weak topology weakly compact weakly sequentially compact zero