## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2141

Let { n } and { vn } be as in Definition 7 , and let { în } and { ön } be

closed sets in S ( T ) such that ûn Sõ , Ùn Sõ ' , and x = lim { E ( F1 ) < + E ( ũ , z } ,

& ef . Since the

Let { n } and { vn } be as in Definition 7 , and let { în } and { ön } be

**sequences**ofclosed sets in S ( T ) such that ûn Sõ , Ùn Sõ ' , and x = lim { E ( F1 ) < + E ( ũ , z } ,

& ef . Since the

**sequence**{ E ( vn ) . + E ( un ) } is strongly convergent , it is ...Page 2197

... and this contradiction completes the proof . Q . E . D . 4 LEMMA . Let B be a

complete ( o - complete ) Boolean algebra of projections in the B - space X and

let { Ex } be a monotone generalized

Then ...

... and this contradiction completes the proof . Q . E . D . 4 LEMMA . Let B be a

complete ( o - complete ) Boolean algebra of projections in the B - space X and

let { Ex } be a monotone generalized

**sequence**( a monotone**sequence**) in B .Then ...

Page 2218

If a generalized

projections in a B - space converges weakly to a projection , then it converges

strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

If a generalized

**sequence**of projections in a o - complete Boolean algebra ofprojections in a B - space converges weakly to a projection , then it converges

strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the case ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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