## Linear Operators, Part 1 |

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Page 293

lim s, f(s)a(ds) = 0 is uniform for f in K. Assume now that K is

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

...

lim s, f(s)a(ds) = 0 is uniform for f in K. Assume now that K is

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

...

Page 294

Since there exists an e > 0 such that for each n there is some f, e K such that soloi

,(s) u(ds) = e, we can find a subset A, of E, such that s.s.o.),(ds) - e.g. Since K is

...

Since there exists an e > 0 such that for each n there is some f, e K such that soloi

,(s) u(ds) = e, we can find a subset A, of E, such that s.s.o.),(ds) - e.g. Since K is

**weakly sequentially compact**, a subsequence of {fi} converges weakly, and it may...

Page 314

Q.E.D. 12 THEOREM. A subset K of bass, 2) is

and only if there exists a non-negative u in bass, 2) such that lim Å(E) = 0 a(E)—-

0 uniformly for Že K. PRoof. Let K C bass, 2) be

...

Q.E.D. 12 THEOREM. A subset K of bass, 2) is

**weakly sequentially compact**ifand only if there exists a non-negative u in bass, 2) such that lim Å(E) = 0 a(E)—-

0 uniformly for Že K. PRoof. Let K C bass, 2) be

**weakly sequentially compact**and...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero