## Linear Operators: Spectral theory |

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Page 970

If Xe denotes the characteristic function of the set e in E , and if f is in L2 ( R ) ,

then Xef is in L ( R ) L2 ( R ) and f is the

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

L ( M ) of ...

If Xe denotes the characteristic function of the set e in E , and if f is in L2 ( R ) ,

then Xef is in L ( R ) L2 ( R ) and f is the

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

**limit**in the norm ofL ( M ) of ...

Page 1124

That is , Q ( E ) = Q ( E ) implies E = E . Similarly , 4 ( E ) = Q ( E ) implies E s E . If

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

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

That is , Q ( E ) = Q ( E ) implies E = E . Similarly , 4 ( E ) = Q ( E ) implies E s E . If

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

**limit**9 ( E ) , then it follows from whatwe have already proved that En is an increasing sequence of projections and E ...

Page 1129

Thus E ( e ) is the strong

from Theorem X . 2 . 1 that Om belongs to the algebra A , so that om is a

linear combinations of products of the operators E ; . Since the projections E ;

form ...

Thus E ( e ) is the strong

**limit**of the operators Om . On the other hand , it followsfrom Theorem X . 2 . 1 that Om belongs to the algebra A , so that om is a

**limit**oflinear combinations of products of the operators E ; . Since the projections E ;

form ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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