## Linear Operators: General theory |

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Page 403

It

It

**follows**from the rotational invariance of un that Un may be regarded as a measure defined intrinsically in any n - dimensional real Hilbert space H , without reference to any particular coordinate system in that space .Page 576

It

It

**follows**from Theorem 10 that the map o → E ( 0 ) is a homeomorphism . To verify that it is an isomorphism , it will suffice to show that E ( q ) = 0 only when o is the void set $ . Now if E ( 0 ) = 0 , then X , = { 0 } and o ( T ) .Page 689

The desired conclusion

The desired conclusion

**follows**from Corollary II.3.13 . Q.E.D. 3 COROLLARY . In a reflexive space , the averages A ( a ) are strongly convergent if they are bounded and if T ( n ) / n converges to zero strongly . PROOF .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

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