## Linear Operators: Spectral theory |

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Page 970

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

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

L ( M ) of the generalized sequence { t ( Xef ) } . Equivalently , we write If = lim [ x ...

... then Zef is in Ly ( R ) n 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 ofL ( M ) of the generalized sequence { t ( Xef ) } . Equivalently , we write If = lim [ x ...

Page 976

Nelson Dunford, Jacob T. Schwartz. By using the form of the characters on RTM

the Plancherel theorem may be given a more concrete formulation in the present

case also . It is easily seen that this theorem asserts that if f is in L ( R ” ) , the

...

Nelson Dunford, Jacob T. Schwartz. By using the form of the characters on RTM

the Plancherel theorem may be given a more concrete formulation in the present

case also . It is easily seen that this theorem asserts that if f is in L ( R ” ) , the

**limit**...

Page 1124

Thus , it follows as above that Em = E . This proves that if y ( En ) is increasing

with

way that if P ( En ) is decreasing with the

E ...

Thus , it follows as above that Em = E . This proves that if y ( En ) is increasing

with

**limit**( E ) , then E , has the strong**limit**E . We may show in exactly the sameway that if P ( En ) is decreasing with the

**limit**q ( E ) , then En has the strong**limit**E ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 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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