## Linear Operators: Spectral theory |

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Page 868

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

I is isometrically isomorphic to the

maximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

I is isometrically isomorphic to the

**field**of complex numbers if and only if I ismaximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

Page 1048

Each point æ in E may be written uniquely as x = rw , where r e R , WES , and the

mapping [ r , w ] + x = rw is evidently a homeomorphism of RXS onto Em . Thus

the o -

Each point æ in E may be written uniquely as x = rw , where r e R , WES , and the

mapping [ r , w ] + x = rw is evidently a homeomorphism of RXS onto Em . Thus

the o -

**field**By of Borel subsets of EM is the product o -**field**of the o -**field**BR of ...Page 1153

Thus the product group has a Haar measure 2 ( 2 ) defined on its Borel

. It is natural to expect that the product measure à x a coincides , up to a constant

multiple , with 7 ( 2 ) . This fact will be established in Theorem 7 . RXR 6 ...

Thus the product group has a Haar measure 2 ( 2 ) defined on its Borel

**field**{ ( 2 ). It is natural to expect that the product measure à x a coincides , up to a constant

multiple , with 7 ( 2 ) . This fact will be established in Theorem 7 . RXR 6 ...

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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 |

### Common terms and phrases

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