## Linear Operators: General theory |

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Page 183

it

Conversely , since ( EU EZ ) = $ ( EU ( E2 ) and $ ( E , E2 ) = ° ( E ) ( E2 ) if Ey , E ,

belong to El , v ( uz , ° ( E ) ) is a nonnegative additive set function defined for Ecz

...

it

**follows**readily from the definition of v ( ug ) that v ( un , 6-1 ( E ) ) z v ( 42 , E ) .Conversely , since ( EU EZ ) = $ ( EU ( E2 ) and $ ( E , E2 ) = ° ( E ) ( E2 ) if Ey , E ,

belong to El , v ( uz , ° ( E ) ) is a nonnegative additive set function defined for Ecz

...

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 Un that Mon 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

that I = Ep + Ep . Further T commutes with Ep and Ey 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 ЕРЕр E , Ep = = = 0 , andthat I = Ep + Ep . Further T commutes with Ep and Ey so this direct sum

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

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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