## Linear Operators: Spectral theory |

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Page 1420

Indeed , let { { n } be a sequence in D ( T1 ( T ) ) . Suppose that { n }

zero in the topology of D ( T1 ( t ) ) . Then , by assumption ( b ) , { { n }

to zero in the topology of D ( T2 ( t + t ' ) ) . Conversely , let { { n }

...

Indeed , let { { n } be a sequence in D ( T1 ( T ) ) . Suppose that { n }

**converges**tozero in the topology of D ( T1 ( t ) ) . Then , by assumption ( b ) , { { n }

**converges**to zero in the topology of D ( T2 ( t + t ' ) ) . Conversely , let { { n }

**converge**to zero...

Page 1436

Let { en } be a bounded sequence of elements of D ( T ) such that { Tgn }

each j , 1 si Sk . Then ħ ; = h ; - & - 1 ( hi ) p ; is in D , and Thi = Thị . Thus { h ; }

Let { en } be a bounded sequence of elements of D ( T ) such that { Tgn }

**converges**. Find a subsequence { & n , } = { h ; } such that x * ( hi )**converges**foreach j , 1 si Sk . Then ħ ; = h ; - & - 1 ( hi ) p ; is in D , and Thi = Thị . Thus { h ; }

**converges**...Page 1664

The Fourier series of an element F in D , ( C )

Proof . It follows from the Definition 37 of the topology in D , ( C ) that it suffices to

show that n ( 29 ' l eil• * Q ( x ) dx JC

q ...

The Fourier series of an element F in D , ( C )

**converges**unconditionally to F .Proof . It follows from the Definition 37 of the topology in D , ( C ) that it suffices to

show that n ( 29 ' l eil• * Q ( x ) dx JC

**converges**unconditionally to F ( q ) for eachq ...

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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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