## Linear Operators: General theory |

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Page 306

2 ) the formula u ( E ) = Set ( s ) a ( ds ) establishes an isometric isomorphism

between

corresponding to K is sequentially weakly compact . Theorem 8 . 9 shows that the

...

2 ) the formula u ( E ) = Set ( s ) a ( ds ) establishes an isometric isomorphism

between

**ca**(**S**, £ , 2 ) and L ( S , E , 2 ) . Thus the set K ' in L ( S , E , 2 )corresponding to K is sequentially weakly compact . Theorem 8 . 9 shows that the

...

Page 308

establishes an equivalence between

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

the hypothesis of Theorem 2 , à may be chosen so that 2 ( E ) S sup su ( E ) ] ...

establishes an equivalence between

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

the hypothesis of Theorem 2 , à may be chosen so that 2 ( E ) S sup su ( E ) ] ...

Page 499

To prove the final statement of the theorem consider the subspace

of

general Radon - Nikodım theorem ( III . 10 . 7 ) , and Theorem III . 2 . 20 ( a ) , the

...

To prove the final statement of the theorem consider the subspace

**ca**(**S**, E , u )of

**ca**(**S**, E ) which consists of all u - continuous functions in**ca**(**S**, E ) . By thegeneral Radon - Nikodım theorem ( III . 10 . 7 ) , and Theorem III . 2 . 20 ( a ) , the

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 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 |

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