## Linear Operators: General theory |

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Page 22

This contradiction proves that A is

that A is

. Let p„ be an arbitrary point of A, and let d0 = sup q(p0. p). The number d0 is

finite, ...

This contradiction proves that A is

**sequentially compact**. Conversely, supposethat A is

**sequentially compact**and closed. It will first be shown that A is separable. Let p„ be an arbitrary point of A, and let d0 = sup q(p0. p). The number d0 is

finite, ...

Page 314

A subset K of ba(S, 27) is weakly

non-negative /< in ba(S, 27) such that lim k(E) = 0 uniformly for A e K. Proof. Let

K Q ba(S, 27) be weakly

A subset K of ba(S, 27) is weakly

**sequentially compact**if and only if there exists anon-negative /< in ba(S, 27) such that lim k(E) = 0 uniformly for A e K. Proof. Let

K Q ba(S, 27) be weakly

**sequentially compact**and let V = UT be the isometric ...Page 511

If A is compact in the strong operator topology, so is the strong operator closure of

co(^4). 4 If a set A C B(£. SJ)) is

topology, its weak operator closure is compact in the weak operator topology.

If A is compact in the strong operator topology, so is the strong operator closure of

co(^4). 4 If a set A C B(£. SJ)) is

**sequentially compact**in the weak operatortopology, its weak operator closure is compact in the weak operator topology.

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact