## Linear Operators: Spectral theory |

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Page 970

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

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

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

x ...

... then Xef is in Lj ( R ) Ly ( 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 ofL2 ( Mo ) of the generalized sequence { t ( Xef ) } . Equivalently , we write tf = lim [

x ...

Page 1124

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

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

SE . If E . is the strong

If Ex , 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 ,

SE . If E . is the strong

**limit**of En , then E . SE and Q ( E ) = Q ( E ) . Thus , it ...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

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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