## 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 {a, o, e

A} is any

Since there is a

**complete**orthonormal set containing the element wo we clearlyhave |T| < |T|*+e and hence |T| < |T||. Q.E.D. 3 CoRollary. If T is in HS and {a, o, e

A} is any

**complete**orthonormal set in Š), then |T|| = { X (Tr., a,)|*}}. a, A 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

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero