## Linear Operators: General theory |

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Page 183

it

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 additive set function defined for E ...

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 additive set function defined for E ...

Page 576

It

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 ) = $ . It

It

**follows**from Theorem 10 that the map o → E ( 0 ) is a homeomorphism . Toverify 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 ) = $ . It

**follows**from ...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

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**from ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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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