## Linear Operators: General theory |

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Page 293

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

integrals Sel ( s ) u ( ds ) are not countably additive uniformly with respect to f in K

there is a ...

lim , ( s ) u ( 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 theintegrals Sel ( s ) u ( ds ) are not countably additive uniformly with respect to f in K

there is a ...

Page 294

... 2 E , we can find a subset An of En such that Ja tu ( s ) u ( ds ) 2 € / 2 . Since K

is weakly

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

... 2 E , we can find a subset An of En such that Ja tu ( s ) u ( ds ) 2 € / 2 . Since K

is weakly

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

Page 314

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

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

unifarmly for à e K. PROOF . Let K C ba ( S , ) be weakly

and let ...

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

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

unifarmly for à e K. PROOF . Let K C ba ( S , ) be weakly

**sequentially**compactand let ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero