Namespaces
Variants
Actions

Strong integral

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

An integral of Lebesgue type, either of a function with values in a linear topological space with respect to a scalar-valued measure, or of a scalar-valued function with respect to a vector-valued measure. Here, the limit processes by which the integral is defined are taken in the sense of the strong topology. Examples of strong integrals are:

1) The Bochner integral of a vector-valued function.

2) The Daniell integral, if the values of the integrand belong to a $ \sigma $-complete vector lattice.

3) The integral $ \displaystyle \int_{- \infty}^{\infty} \lambda ~ \mathrm{d}{F_{\lambda}} $, giving the spectral decomposition of a self-adjoint operator acting on a Hilbert space.

For the strong integral of scalar-valued functions with respect to a vector-valued measure, the values of the measure, in many cases, are assumed to belong to a semi-ordered vector space.

References

[1a] N. Dunford and J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).
[1b] N. Dunford and J.T. Schwartz, “Linear operators. Spectral theory”, 2, Interscience (1963).
[2] T.H. Hildebrandt, “Integration in abstract spaces”, Bull. Amer. Math. Soc., 59 (1953), pp. 111–139.

Comments

See also vector measure; strong topology.

References

[a1] J. Diestel and J.J. Uhl, Jr., “Vector measures”, Math. Surveys, 15, Amer. Math. Soc. (1977).
How to Cite This Entry:
Strong integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strong_integral&oldid=38660
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article