## Linear Operators, Part 2 |

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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 Ly ( R ) , L ( 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 Ly ( R ) , L ( 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 ofL ( M ) of ...

Page 1124

Hence ( Exn | 2 = E , x , for each n , so that Exn = EzXn and E = E . That is , Q ( E )

= 4 ( E ) implies E = E . Similarly , Q ( E ) S Q ( E ) implies E s Eq . If En . E are in F

and ( En ) increases to the

Hence ( Exn | 2 = E , x , for each n , so that Exn = EzXn and E = E . That is , Q ( E )

= 4 ( E ) implies E = E . Similarly , Q ( E ) S Q ( E ) implies E s Eq . If En . E are in F

and ( En ) increases to the

**limit**( E ) , then it follows from what we have already ...Page 1129

Thus E ( e ) is the strong

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

combinations of products of the operators E . Since the projections E ; form a ...

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 is a

**limit**of linearcombinations of products of the operators E . Since the projections E ; form a ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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