## Linear Operators: General theory |

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Page 183

it

. Conversely , since $ ( EU EZ ) = $ ( EU $ ( Eg ) and $ ( EEZ ) = $ ( E2 ) ( EZ ) if

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

...

it

**follows**readily from the definition of v ( ug ) that v ( 417 , 6 - 1 ( E ) ) 2 0 ( 42 , E ). Conversely , since $ ( EU EZ ) = $ ( EU $ ( Eg ) and $ ( EEZ ) = $ ( E2 ) ( EZ ) if

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

...

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 $ . Now if E ( 0 ) = 0 , then Xo = { 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 $ . Now if E ( 0 ) = 0 , then Xo = { 0 } and o ( T . ) = $ . It

**follows**from...

Page 689

The desired conclusion

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 . This

The desired conclusion

**follows**from Corollary II . 3 . 13 . Q . E . D . 3COROLLARY . 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 . This

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

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect 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