## Linear Operators, Part 1 |

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Page 271

Hence we have

1 ( so ) < E , feF 09 1SiST which proves the quasi - uniform convergence of f ( sn )

to f ( so ) . Thus ( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) . Let F. { / 1

...

Hence we have

**shown**that there exists ni , . ... , No 2 no such that min \ | ( sn , ) -1 ( so ) < E , feF 09 1SiST which proves the quasi - uniform convergence of f ( sn )

to f ( so ) . Thus ( 3 ) implies ( 4 ) . We now show that ( 4 ) implies ( 5 ) . Let F. { / 1

...

Page 291

the closed subspace L ( E ) = L ( E , E ( E ) , u ) of L ( S , E , u ) consisting of all

functions vanishing outside E. If we can find an feLi ( E ) to which { fr } converges

weakly , we will have

the closed subspace L ( E ) = L ( E , E ( E ) , u ) of L ( S , E , u ) consisting of all

functions vanishing outside E. If we can find an feLi ( E ) to which { fr } converges

weakly , we will have

**shown**that L , is weakly complete . Since { In } is a weak ...Page 684

It was observed in the preceding proof that the limit g ( s ) exists for ualmost all s

and hence it only remains to be

the preceding proof that m ( q - le ) = m ( e ) for e in and thus the mean ergodic ...

It was observed in the preceding proof that the limit g ( s ) exists for ualmost all s

and hence it only remains to be

**shown**that g is u - integrable . It was observed inthe preceding proof that m ( q - le ) = m ( e ) for e in and thus the mean ergodic ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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