## 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 g(p0, p). The number rf0 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 p0 be an arbitrary point of A, and let d0 = sup g(p0, p). The number rf0 is

finite ...

Page 314

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

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

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

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

**sequentially compact**if andonly if there exists a non-negative p 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 = UT be ...Page 511

Then if A is compact in the weak operator topology, so is the weak operator

closure of co(A). HA is compact in the strong operator topology, so is the strong

operator closure of co(^4). 4 If a set A C Z?(3E, ?) ) is

weak ...

Then if A is compact in the weak operator topology, so is the weak operator

closure of co(A). HA is compact in the strong operator topology, so is the strong

operator closure of co(^4). 4 If a set A C Z?(3E, ?) ) is

**sequentially compact**in theweak ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Common terms and phrases

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