## Linear Operators, Part 2 |

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Page 891

scalar function f with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded functions f and so the ... We shall use

the termZ'-

scalar function f with respect to the operator valued set function E. In the present

chapter we shall only integrate bounded functions f and so the ... We shall use

the termZ'-

**simple function**for any function on S having the form (viii) / = 2 W, .Page 892

This shows that if the sequence {fa} of Z-

then the sequence {IS /,,(s)E(ds)} of integrals converges, and its limit depends

only upon f and not upon the particular sequence {fa} used to approximate I. Thus

...

This shows that if the sequence {fa} of Z-

**simple functions**converges in B(S,Z) to fthen the sequence {IS /,,(s)E(ds)} of integrals converges, and its limit depends

only upon f and not upon the particular sequence {fa} used to approximate I. Thus

...

Page 1218

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. ... Since u(R) < 00 every Iu-measurable function f is the limit in

'u-measure of a sequence {fn} of ,u-

The following lemma is, in the case of scalar functions, a well known theorem of

Lusin. 17 LEMMA. ... Since u(R) < 00 every Iu-measurable function f is the limit in

'u-measure of a sequence {fn} of ,u-

**simple functions**. In view of Corollary III.6.8 it ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

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