## Linear Operators: Spectral theory |

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Page 1011

Since there is a

have |T| < |T|*+e and hence |T| < |T||. Q.E.D. 3 CorollARY. If T is in HS and {rx, & e

A} is any

Since there is a

**complete**orthonormal set containing the element ro we clearlyhave |T| < |T|*+e and hence |T| < |T||. Q.E.D. 3 CorollARY. If T is in HS and {rx, & e

A} is any

**complete**orthonormal set in S), then |T|| = { X (Tr., ra)*}}. 2, 6 e A PRoof.Page 1147

CoRoLLARy: If G is a compact topological group satisfying the second axiom of

countability, and G is not a finite set, then any

G is countable. A

CoRoLLARy: If G is a compact topological group satisfying the second axiom of

countability, and G is not a finite set, then any

**complete**set of representations ofG is countable. A

**complete**set of representations of a finite group is finite.Page 1903

(See B-space)

partially ordered space, definition, I.3.9 (8) Completely regular space,

compactification of, IV.6.22 (276), IX.2.16 (872) definition, IV.6.21 (276), IX.2.15 (

872) ...

(See B-space)

**Complete**orthonormal set, in Hilbert space, IV.4.8 (250)**Complete**partially ordered space, definition, I.3.9 (8) Completely regular space,

compactification of, IV.6.22 (276), IX.2.16 (872) definition, IV.6.21 (276), IX.2.15 (

872) ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 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 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