## Linear Operators: General theory |

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Page 311

1 shows that there is an isometric

ba ( S , E ' ) , which is determined by the equation w * XE = 4 ( E ) , E € E . Thus ,

since B ( S , E ) is equivalent to C ( Si ) , ba ( S , E ) is equivalent to rca ( S ...

1 shows that there is an isometric

**isomorphism**x * < > u between B * ( S , E ) andba ( S , E ' ) , which is determined by the equation w * XE = 4 ( E ) , E € E . Thus ,

since B ( S , E ) is equivalent to C ( Si ) , ba ( S , E ) is equivalent to rca ( S ...

Page 312

Let S , be a compact Hausdorff space such that B ( S , E ) is isometrically

, t ( EUF ) = t ...

Let S , be a compact Hausdorff space such that B ( S , E ) is isometrically

**isomorphic**with C ( S ) . ... The correspondence % e →Xe , establishes an**isomorphism**t of the field onto the field E , of all open and closed sets in S , , i . e ., t ( EUF ) = t ...

Page 313

The correspondence U : My → Ma is an isometric

onto ca ( S1 , E2 ) . ( c ) If E , is in & , then v ( un , Eq ) = v ( U ( 47 ) , E ) for all My

in ba ( S , ; ) . Proof . Recalling that t is an

the ...

The correspondence U : My → Ma is an isometric

**isomorphism**of ba ( S1 , Ey )onto ca ( S1 , E2 ) . ( c ) If E , is in & , then v ( un , Eq ) = v ( U ( 47 ) , E ) for all My

in ba ( S , ; ) . Proof . Recalling that t is an

**isomorphism**of onto & , it is clear thatthe ...

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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