## Linear Operators: General theory |

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Page 311

Theorem 5 . 1 shows that there is an

* ( S , E ) and ba ( S , E ' ) , which is determined by the equation * * % £ = u ( E ) ,

E € £ . Thus , since B ( S , E ) is equivalent to C ( S ) , ba ( S , E ) is equivalent to ...

Theorem 5 . 1 shows that there is an

**isometric isomorphism**æ * < > u between B* ( S , E ) and ba ( S , E ' ) , which is determined by the equation * * % £ = u ( E ) ,

E € £ . Thus , since B ( S , E ) is equivalent to C ( S ) , ba ( S , E ) is equivalent to ...

Page 313

Recalling that t is an isomorphism of onto 2 , it is clear that the mapping T is an

TE : ) ) = sup \ u ( E ; ) ] = level , where { E1 , . . . , En } is an arbitrary partition of S ...

Recalling that t is an isomorphism of onto 2 , it is clear that the mapping T is an

**isometric isomorphism**of ba ( S , E ) onto ba ( S1 , E , ) , since Tul = sup Š | ( Tu ) (TE : ) ) = sup \ u ( E ; ) ] = level , where { E1 , . . . , En } is an arbitrary partition of S ...

Page 337

Thus , ba ( S , E ) is isometrically isomorphic with the closed subspace BV ( I ) of

all f e BV ( I ) such that f ( a + ) = 0 . If N is the one ... Thus f + My determines an

Thus , ba ( S , E ) is isometrically isomorphic with the closed subspace BV ( I ) of

all f e BV ( I ) such that f ( a + ) = 0 . If N is the one ... Thus f + My determines an

**isometric isomorphism**between NBV ( I ) and rba ( I , E ) . Using Theorem 9 .### What people are saying - Write a review

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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