## Linear Operators: General theory |

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Page 164

Let C be the set of all u in

chosen from C in such a way that lim , Mn ( S ) = supuec M ( S ) < 0 . Since Mi S E

- 1 Mj , i = 1 , . . . , n , it follows from Corollary 6 that { M , . . . , Mn } has a least ...

Let C be the set of all u in

**ca**(**S**, E ) such that 0 Su Sa . Let Mn , n = 1 , 2 , . . . , bechosen from C in such a way that lim , Mn ( S ) = supuec M ( S ) < 0 . Since Mi S E

- 1 Mj , i = 1 , . . . , n , it follows from Corollary 6 that { M , . . . , Mn } has a least ...

Page 306

The functions Mm are all continuous with respect to the measure defined by 1 v (

un , E ) 2 ( E ) = ani Ee E , n = 1 2 " 1 + v ( un , E ) ' and thus all belong to the

subspace

The functions Mm are all continuous with respect to the measure defined by 1 v (

un , E ) 2 ( E ) = ani Ee E , n = 1 2 " 1 + v ( un , E ) ' and thus all belong to the

subspace

**ca**(**S**, E , ) consisting of all 2 - continuous functions in**ca**(**S**, E ) .Page 308

establishes an equivalence between

present theorem follows from Corollary 8 . 11 . Q . E . D . 3 COROLLARY . Under

the hypothesis of Theorem 2 , 2 may be chosen so that 2 ( E ) 3 sup lu ( E ) ] , E ...

establishes an equivalence between

**ca**(**S**, E , 2 ) and L ( S , E , 2 ) and thus thepresent theorem follows from Corollary 8 . 11 . Q . E . D . 3 COROLLARY . Under

the hypothesis of Theorem 2 , 2 may be chosen so that 2 ( E ) 3 sup lu ( E ) ] , E ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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

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