## Linear Operators: General theory |

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Page 22

This contradiction proves that A is

that A is

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

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 p0 be an arbitrary point of A, and let d0 = sup Q{p0. p). The number d0 is ...

Page 314

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly

only if there exists a non-negative fj, in ba(S, E) such that lim X(E) = 0 uniformly

for XeK. Proof. Let K Q ba(S, E) be weakly

be ...

Q.E.D. 12 Theorem. A subset K of ba(S, E) is weakly

**sequentially compact**if andonly if there exists a non-negative fj, in ba(S, E) such that lim X(E) = 0 uniformly

for XeK. Proof. Let K Q ba(S, E) be weakly

**sequentially compact**and let V = UTbe ...

Page 511

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

co(A). 4 If a set ACB(£, ?)) is

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(A). 4 If a set ACB(£, ?)) is

**sequentially compact**in the weak operator topology,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 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