## Linear Operators: Spectral theory |

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Page 891

scalar

chapter we shall only integrate bounded

discussion of the integral will be restricted to that case . Let Ebe a field of subsets

of a set ...

scalar

**function**| with respect to the operator valued set**function**E . In the presentchapter we shall only integrate bounded

**functions**| and so the followingdiscussion of the integral will be restricted to that case . Let Ebe a field of subsets

of a set ...

Page 1178

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

, of scalar - valued

which maps the vector - valued

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

, of scalar - valued

**functions**into the space L ... Let M be the mapping in L ( 12 )which maps the vector - valued

**function**whose nth component has the Fourier ...Page 1922

1 ( 418 ) Total variation of a

97 ) . ( See also Variation ) Totally ordered set , 1 . 2 . 2 ( 4 ) Trace , of a finite

matrix , VI . 9 . 28 ( 515 ) , XI . 6 . 8 ( 1016 ) of a matrix , definition , VI . 9 . 28 ( 515

) of ...

1 ( 418 ) Total variation of a

**function**, III . 5 . 15 ( 140 ) of a set**function**, III . 1 . 4 (97 ) . ( See also Variation ) Totally ordered set , 1 . 2 . 2 ( 4 ) Trace , of a finite

matrix , VI . 9 . 28 ( 515 ) , XI . 6 . 8 ( 1016 ) of a matrix , definition , VI . 9 . 28 ( 515

) of ...

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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 |

### Common terms and phrases

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