## 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

Since K is weakly

weakly , and it may be assumed without loss of generality that { / n } itself

converges weakly . By Theorem 7 , { Sein ( s ) u ( ds ) } converges for each Ee E .

Let 1 ...

Since K is weakly

**sequentially**compact , a subsequence of { { n } convergesweakly , and it may be assumed without loss of generality that { / n } itself

converges weakly . By Theorem 7 , { Sein ( s ) u ( ds ) } converges for each Ee E .

Let 1 ...

Page 314

A subset K of ba ( S , E ) is weakly

a non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 H ( E ) 0 unifarmly for a €

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

A subset K of ba ( S , E ) is weakly

**sequentially**compact if and only if there existsa non - negative u in ba ( S , E ) such that lim 2 ( E ) = 0 H ( E ) 0 unifarmly for a €

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

**sequentially**compact and let V = UT be ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero