## Linear Operators: General theory |

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Page 293

lim L f {')/*(*») = 0 is uniform for / in K. Assume now that K is

ds) are not countably additive uniformly with respect to / in K there is a number e

...

lim L f {')/*(*») = 0 is uniform for / in K. Assume now that K is

**weakly sequentially****compact**. It follows from Lemma II.3.27 that K is bounded. If the integrals J" Ff(s)/j.(ds) are not countably additive uniformly with respect to / in K there is a number e

...

Page 294

Since K is

and it may be assumed without loss of generality that {/„} itself converges weakly.

By Theorem 7, {J Ef„(s)fi(ds)} converges for each EeE. Let 1 K(E) K(E) = jE ...

Since K is

**weakly sequentially compact**, a subsequence of {/„} converges weakly,and it may be assumed without loss of generality that {/„} itself converges weakly.

By Theorem 7, {J Ef„(s)fi(ds)} converges for each EeE. Let 1 K(E) K(E) = jE ...

Page 314

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

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

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 ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 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

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