## Linear Operators: Spectral theory |

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scalar

scalar

**functions**with respect to the operator valued set**function**E. In the present chapter we shall only integrate bounded**functions**f and so the following discussion of the integral will be restricted to that case .Page 1178

It is plain from Plancherel's theorem that X is a bounded mapping of the space L , of scalar - valued

It is plain from Plancherel's theorem that X is a bounded mapping of the space L , of scalar - valued

**functions**into the ... Let M be the mapping in L , ( 12 ) which maps the vector - valued**function**whose nth component has the Fourier ...Page 1922

... XII.5.1 ( 1240 ) Urysohn theorems , metrization , 1.6.19 ( 24 ) for normal spaces , 1.5.2 ( 15 ) V ( 150 ) , III.9.45 , IV.10.9 ( 325 ). Total space of functionals , definition , V.3.1 ( 418 ) Total variation of a

... XII.5.1 ( 1240 ) Urysohn theorems , metrization , 1.6.19 ( 24 ) for normal spaces , 1.5.2 ( 15 ) V ( 150 ) , III.9.45 , IV.10.9 ( 325 ). Total space of functionals , definition , V.3.1 ( 418 ) Total variation of a

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

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### Common terms and phrases

additive Akad algebra Amer analytic applied 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 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 satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero