## Linear Operators: Spectral theory |

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Page 1332

To prove (c), we have only to show that the set of

Since for each element f of our Hilbert space we have f = X E(2)f, Ae or (T) and

since we have observed above that E(2)f is an

To prove (c), we have only to show that the set of

**eigenfunctions**of T is complete.Since for each element f of our Hilbert space we have f = X E(2)f, Ae or (T) and

since we have observed above that E(2)f is an

**eigenfunction**of T, this is obvious.Page 1383

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of

conditions A and C, the unique solution of tao = Ao satisfying the boundary

condition ...

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of

**eigenfunctions**being complete. With boundaryconditions A and C, the unique solution of tao = Ao satisfying the boundary

condition ...

Page 1617

in the study of convergence in L, The main results of these papers are the

following: (1) Haar [3] proved that for the Sturm-Liouville

by the impositions of separated boundary conditions (a) there exist continuous ...

in the study of convergence in L, The main results of these papers are the

following: (1) Haar [3] proved that for the Sturm-Liouville

**eigenfunctions**obtainedby the impositions of separated boundary conditions (a) there exist continuous ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### 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