## 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 o(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 o(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 u in ba(S, E) such that lim X(E) = 0 uniformly for

X e K. 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 aridonly if there exists a non-negative u in ba(S, E) such that lim X(E) = 0 uniformly for

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

**sequentially compact**and let V = UT be ...Page 434

We thus conclude that K is weakly

convex set and therefore (3.13) X*-closed, the preceding theorem applies to

show that K is X*-compact. Q.E.D. 3 Theorem. The weak topology of a weakly ...

We thus conclude that K is weakly

**sequentially compact**, but since K is a closedconvex set and therefore (3.13) X*-closed, the preceding theorem applies to

show that K is X*-compact. Q.E.D. 3 Theorem. The weak topology of a weakly ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

79 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero