## Linear Operators, Part 1 |

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Page 183

it

Conversely , since $ ( EU Ey ) = ( EU ( E ) and $ ( E , E2 ) = $ ( E4 ) $ ( E2 ) if Eį , E

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

it

**follows**readily from the definition of v ( ug ) that v ( un , 6-1 ( E ) ) z v ( uz , E ) .Conversely , since $ ( EU Ey ) = ( EU ( E ) and $ ( E , E2 ) = $ ( E4 ) $ ( E2 ) if Eį , E

, belong to El , 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 Mn 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 689

The desired conclusion

a reflexive space , the averages A ( Q ) are strongly convergent if they are

bounded and if T ( n ) / n converges to zero strongly . Proof . This

Theorem ...

The desired conclusion

**follows**from Corollary II.3.13 . Q.E.D. 3 COROLLARY . Ina reflexive space , the averages A ( Q ) are strongly convergent if they are

bounded and if T ( n ) / n converges to zero strongly . Proof . This

**follows**fromTheorem ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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