## Linear Operators, Part 2 |

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Page 984

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

(R, .9¢?,,u) which vanish outside of compact sets is

The set of functions f in L1(R) for which f vanishes in a neighborhood of infinity is

**dense**in L1(R). Pnoor. It follows from Lemma 3.6 that the set of all functions in L2(R, .9¢?,,u) which vanish outside of compact sets is

**dense**in this space, and ...Page 1246

We may also regard A as a mapping from the

space $1. In this case A is still continuous, for |A.z'|f = (A1, Aw), = (A213 as), = (Aw

, .2), .2: e §D(T), and, by the inequalities above, (Am, ac) § |A.r| |.v| § ]Ax|1|.r|, ...

We may also regard A as a mapping from the

**dense**subspace $(T) of Q) into thespace $1. In this case A is still continuous, for |A.z'|f = (A1, Aw), = (A213 as), = (Aw

, .2), .2: e §D(T), and, by the inequalities above, (Am, ac) § |A.r| |.v| § ]Ax|1|.r|, ...

Page 1905

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489)

L,, 1ll.9.l7 (170), 1v.1-1.19 (295) density of simple functions in L,, 1 § P < <1) ...

(2)

**Dense**convex sets, V.7.2'l (487)**Dense**linear manifolds, V.7.40—41 (438-489)

**Dense**set, definition, 1.6.11 (21) density of continuous functions in TM andL,, 1ll.9.l7 (170), 1v.1-1.19 (295) density of simple functions in L,, 1 § P < <1) ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Paoor partial differential operator Pnoor positive preceding lemma Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero