## Linear Operators: General theory |

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Page 183

it

it

**follows**readily from the definition of v ( uz ) that v / m , 8-1 ( E ) ) 2 v ( l2 , E ) . Conversely , since $ ( EU Ey ) = 4 ( EU 4 ( E ) and $ ( EEZ ) = $ ( E ) ( E2 ) if Ez , E , belong to Ej , v ( la , 4 ( E ) ) is a nonnegative ...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 ( 0 ) = 0 only when o is the void set 0. 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 ( Q ) 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 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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### Common terms and phrases

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