## 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 field of

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 872

and each x in X define x ( a ) = lim Pn ( a ) where { Pn } is a sequence of

polynomials with [ Pn ( z ) — 21 + 0 . The number x ( ) is clearly independent of

the ...

**complex**variable that { Pn ( 2 ) } also converges uniformly on G . For each 1 in Gand each x in X define x ( a ) = lim Pn ( a ) where { Pn } is a sequence of

polynomials with [ Pn ( z ) — 21 + 0 . The number x ( ) is clearly independent of

the ...

Page 887

... Notions The spectral theorem to be proved in this chapter will introduce a

theory which parallels in Hilbert space the thcory in n - dimensional unitary space

associated with the classical reduction of a finite normal matrix of

numbers .

... Notions The spectral theorem to be proved in this chapter will introduce a

theory which parallels in Hilbert space the thcory in n - dimensional unitary space

associated with the classical reduction of a finite normal matrix of

**complex**numbers .

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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