Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.3, p. 27   | 1 | 2 |

Section Integration, [L^{p}] spaces

G. Bricognea

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Integration, [L^{p}] spaces

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The Riemann integral used in elementary calculus suffers from the drawback that vector spaces of Riemann-integrable functions over [{\bb R}^{n}] are not complete for the topology of convergence in the mean: a Cauchy sequence of integrable functions may converge to a non-integrable function.

To obtain the property of completeness, which is fundamental in functional analysis, it was necessary to extend the notion of integral. This was accomplished by Lebesgue [see Berberian (1962)[link], Dieudonné (1970)[link], or Chapter 1 of Dym & McKean (1972)[link] and the references therein, or Chapter 9 of Sprecher (1970)[link]], and entailed identifying functions which differed only on a subset of zero measure in [{\bb R}^{n}] (such functions are said to be equal `almost everywhere'). The vector spaces [L^{p} ({\bb R}^{n})] consisting of function classes f modulo this identification for which [\|{\bf f}\|_{p} = \left({\textstyle\int\limits_{{\bb R}^{n}}} |\;f ({\bf x}) |^{p}\ {\rm d}^{n} {\bf x}\right)^{1/p} \;\lt\; \infty] are then complete for the topology induced by the norm [\|.\|_{p}]: the limit of every Cauchy sequence of functions in [L^{p}] is itself a function in [L^{p}] (Riesz–Fischer theorem).

The space [L^{1} ({\bb R}^{n})] consists of those function classes f such that [\|\;f \|_{1} = {\textstyle\int\limits_{{\bb R}^{n}}} |\;f ({\bf x})|\;\hbox{d}^{n} {\bf x} \;\lt\; \infty] which are called summable or absolutely integrable. The convolution product: [\eqalign{(\;f * g) ({\bf x}) &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf y}) g({\bf x} - {\bf y})\;\hbox{d}^{n} {\bf y}\cr &= {\textstyle\int\limits_{{\bb R}^{n}}} f({\bf x} - {\bf y}) g ({\bf y})\;\hbox{d}^{n} {\bf y} = (g * f) ({\bf x})}] is well defined; combined with the vector space structure of [L^{1}], it makes [L^{1}] into a (commutative) convolution algebra. However, this algebra has no unit element: there is no [f \in L^{1}] such that [f * g = g] for all [g \in L^{1}]; it has only approximate units, i.e. sequences [(f_{\nu })] such that [f_{\nu } * g] tends to g in the [L^{1}] topology as [\nu \rightarrow \infty]. This is one of the starting points of distribution theory.

The space [L^{2} ({\bb R}^{n})] of square-integrable functions can be endowed with a scalar product [(\;f, g) = {\textstyle\int\limits_{{\bb R}^{n}}} \overline{f({\bf x})} g({\bf x})\;\hbox{d}^{n} {\bf x}] which makes it into a Hilbert space. The Cauchy–Schwarz inequality [|(\;f, g)| \leq [(\;f, f) (g, g)]^{1/2}] generalizes the fact that the absolute value of the cosine of an angle is less than or equal to 1.

The space [L^{\infty} ({\bb R}^{n})] is defined as the space of functions f such that [\|\;f \|_{\infty} = \lim\limits_{p \rightarrow \infty} \|\;f \|_{p} = \lim\limits_{p \rightarrow \infty} \left({\textstyle\int\limits_{{\bb R}^{n}}} |\; f({\bf x}) |^{p} \;\hbox{d}^{n} {\bf x}\right)^{1/p} \;\lt\; \infty.] The quantity [\|\;f \|_{\infty}] is called the `essential sup norm' of f, as it is the smallest positive number which [|\;f({\bf x})|] exceeds only on a subset of zero measure in [{\bb R}^{n}]. A function [f \in L^{\infty}] is called essentially bounded.


Berberian, S. K. (1962). Measure and integration. New York: Macmillan. [Reprinted by Chelsea, New York, 1965.]
Dieudonné, J. (1970). Treatise on analysis, Vol. II. New York and London: Academic Press.
Dym, H. & McKean, H. P. (1972). Fourier series and integrals. New York and London: Academic Press.
Sprecher, D. A. (1970). Elements of real analysis. New York: Academic Press. [Reprinted by Dover Publications, New York, 1987.]

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