An approach to F. Riesz representation Theorem

Rafael del Rio; Asaf L. Franco; Jose A. Lara

doi:10.4067/S0719-06462018000200001

DOI:

https://doi.org/10.4067/S0719-06462018000200001

Abstract

In this note we give a direct proof of the F. Riesz representation theorem which characterizes the linear functionals acting on the vector space of continuous functions defined on a set K. Our start point is the original formulation of Riesz where K is a closed interval. Using elementary measure theory, we give a proof for the case K is an arbitrary compact set of real numbers. Our proof avoids complicated arguments commonly used in the description of such functionals.

Keywords

Riesz representation theorem , positive linear functionals , Riemann Stieltjes integral

Rafael del Rio Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México C.P. 04510, CDMX, México.
Asaf L. Franco Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México C.P. 04510, CDMX, México.
Jose A. Lara Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México C.P. 04510, CDMX, México.

Pages: 01–12
Date Published: 2018-07-31
Vol. 20 No. 2 (2018)

Bartle, Robert G. The elements of integration. John Wiley & Sons, Inc., New York-London- Sydney 1966 x+129 pp.

V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007.

D. Cohn, Measure theory, secon ed., Birkh Ìˆauser, Boston, Mass. 2013.

Doob, J. L. Measure theory. Graduate Texts in Mathematics, 143. Springer-Verlag, New York, 1994. xii+210 pp. ISBN: 0-387-94055-3

Gerald B. Folland, Real analysis, second ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1999, Modern techniques and their applications, A Wiley-Interscience Publication.

Gray, J. D. The shaping of the Riesz representation theorem: a chapter in the history of analysis. Arch. Hist. Exact Sci. 31 (1984), no. 2, 127–187.

D. G. Hartig, The Riesz Representation Theorem Revisited, Amer. Math. Monthly 90 No. 4 (1983), pp. 277–280

E. Helly, U Ìˆber lineare Funktionaloperationen, Wien Ber. 121 (1912), 265–297.

Shizuo Kakutani, Concrete representation of abstract (M)-spaces. (A characterization of the space of continuous functions.), Ann. of Math. (2) 42 (1941), 994–1024.

Erwin Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978.

A. Markoff, On mean values and exterior densities, Mat. Sbornik 4 (46) (1938), no. 1, 165–191.

Johann Radon, Gesammelte Abhandlungen. Band 1, Verlag der O Ìˆ sterreichischen Akademie der Wissenschaften, Vienna; Birkh Ìˆauser Verlag, Basel, 1987, With a foreword by Otto Hittmair, Edited and with a preface by Peter Manfred Gruber, Edmund Hlawka, Wilfried No Ìˆbauer and Leopold Schmetterer.

Michael Reed and Barry Simon, Methods of modern mathematical physics. I, second ed., Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980, Functional analysis.

F. Riesz, Sur les op Ìerations fonctionnelles lin Ìeares, Comptes Rendus Acad. Sci. Paris 149 (1909), 974–977.

Demonstration nouvelle d‘un th Ìeor`eme concernant les op Ìerations, Annales Ecole Norm. Sup. 31 (1914), 9–14.

Fr Ìed Ìeric Riesz, Sur la repr Ìesentation des op Ìerations fonctionnelles lin Ìeaires par des int Ìegrales de Stieltjes, Comm. S Ìem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (1952), no. Tome Supplementaire, 181–185.

Frigyes Riesz and B Ìela Sz.-Nagy, Functional analysis, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1990, Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original.

H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963.

Walter Rudin, Real and Complex Analysis 3rd. ed., McGraw-Hill, Inc., 1987, New York, NY, USA.

S. Saks, Integration in abstract metric spaces, Duke Math. J. 4 (1938), no. 2, 408–411.

Stanislô°€aw Saks, Theory of the integral, Second revised edition. English translation by L. C. Young. With two additional notes by Stefan Banach, Dover Publications, Inc., New York, 1964.

Martin Schechter, Principles of functional analysis, second ed., Graduate Studies in Mathe- matics, vol. 36, American Mathematical Society, Providence, RI, 2002.

Barry Simon, Real analysis, A Comprehensive Course in Analysis, Part 1, American Mathe- matical Society, Providence, RI, 2015, With a 68 page companion booklet.