## Linear Operators: General theory |

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Page 183

it

Conversely , since ( EU E2 ) = $ ( EU ( E ) and $ ( E , E2 ) = 4 ( E ) * ( E2 ) if E2 , E

, belong to Ły , v ( uz , $ ( E ) ) is a nonnegative additive set function defined for ...

it

**follows**readily from the definition of v ( ug ) that v ( un , 9 - 1 ( E ) ) 2 v ( u2 , E ) .Conversely , since ( EU E2 ) = $ ( EU ( E ) and $ ( E , E2 ) = 4 ( E ) * ( E2 ) if E2 , E

, belong to Ły , v ( uz , $ ( E ) ) is a nonnegative additive set function defined for ...

Page 403

It

measure defined intrinsically in any n - dimensional real Hilbert space H , without

reference to any particular coordinate system in that space . This measure will be

...

It

**follows**from the rotational invariance of win that Min may be regarded as ameasure defined intrinsically in any n - dimensional real Hilbert space H , without

reference to any particular coordinate system in that space . This measure will be

...

Page 714

It

I = Ep + Ep . Further T commutes with Ep and Ej so this direct sum decomposition

is into subspaces invariant under T . Statements ( a ) and ( b )

It

**follows**from its definition that Ep is a projection , that E Ep = EyEp = 0 , and thatI = Ep + Ep . Further T commutes with Ep and Ej so this direct sum decomposition

is into subspaces invariant under T . Statements ( a ) and ( b )

**follow**from the ...### 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