## Linear Operators: Spectral theory |

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Page 970

... then Xef is in Ly ( R ) L2 ( R ) and f is the

generalized sequence { Xef } . Hence , by Theorem 9 , tf is the

L2 ( M ) of the generalized sequence { t ( Xef ) } . Equivalently , we write tf lim [ e ,

• ] f ...

... then Xef is in Ly ( R ) L2 ( R ) and f is the

**limit**in the norm of L2 ( R ) of thegeneralized sequence { Xef } . Hence , by Theorem 9 , tf is the

**limit**in the norm ofL2 ( M ) of the generalized sequence { t ( Xef ) } . Equivalently , we write tf lim [ e ,

• ] f ...

Page 1124

That is , Q ( E ) = Q ( E ) implies E = E. Similarly , y ( E ) SQ ( E ) implies ESE . If En

, E are in F and q ( En ) increases to the

have already proved that En is an increasing sequence of projections and En > E

.

That is , Q ( E ) = Q ( E ) implies E = E. Similarly , y ( E ) SQ ( E ) implies ESE . If En

, E are in F and q ( En ) increases to the

**limit**( E ) , then it follows from what wehave already proved that En is an increasing sequence of projections and En > E

.

Page 1699

by Lemma 3.22 , defe is the

functions in CO ( L ) . Putting g , ( x ) = 0 for x in CE - L , it follows from Definition

3.15 that PeF , is the

by Lemma 3.22 , defe is the

**limit**in the norm of H ( P ) ( L ) of a sequence { gj } offunctions in CO ( L ) . Putting g , ( x ) = 0 for x in CE - L , it follows from Definition

3.15 that PeF , is the

**limit**in the norm of H ( P ) ( Ce ) of the sequence { g ; } of ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

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