## Linear Operators, Part 2 |

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Page 1061

a1 g K,,|f],,., by Theorem IV.8.1, the Hahn-Banach theorem (II.8.14-) and H6lder's

the basic

a1 g K,,|f],,., by Theorem IV.8.1, the Hahn-Banach theorem (II.8.14-) and H6lder's

**inequality**, and the theorem is proved for all p, 1 < p < 00. Q.E.D. Having provedthe basic

**inequality**of M. Riesz, we now proceed to prove the full**inequality**of ...Page 1105

\Ve now pause to sharpen another of the

continuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

Lemma ...

\Ve now pause to sharpen another of the

**inequalities**of Lemma 9. ... thecontinuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

**inequality**ofLemma ...

Page 1774

The above

follows from the postulates for Q that the Schwarz

y is zero. Hence suppose that .1: qé 0 qé y. For an arbitrary complex number oi 0

g ...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first. Itfollows from the postulates for Q that the Schwarz

**inequality**is valid if either .2: ory is zero. Hence suppose that .1: qé 0 qé y. For an arbitrary complex number oi 0

g ...

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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