## Linear Operators: Spectral theory |

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Page 1420

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

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

to zero in the topology of D ( T1 ( + t ' ) ) . Conversely , let { In }

...

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

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

**converges**to zero in the topology of D ( T1 ( + t ' ) ) . Conversely , let { In }

**converge**to zero in...

Page 1436

Let { 8n } be a bounded sequence of elements of D ( T ) such that { Tgn }

each j , 1 si s k . Then h ; = h ; - * _ * ( hi ) , is in D , and Tħ ; = Thi . Thus { ħ ; }

Let { 8n } be a bounded sequence of elements of D ( T ) such that { Tgn }

**converges**. Find a subsequence { gn ; } = { h ; } such that x * ( h ; )**converges**foreach j , 1 si s k . Then h ; = h ; - * _ * ( hi ) , is in D , and Tħ ; = Thi . Thus { ħ ; }

**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 ( 27 ) - * { F15 est * 2 plader JC

...

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 ( 27 ) - * { F15 est * 2 plader JC

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

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero