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Abstract and Applied Analysis
Volume 2014, Article ID 609525, 26 pages
Research Article

Function Spaces with Bounded Means and Their Continuous Functionals

1Department of Mathematics, University of Rome “Tor Vergata”, Via Ricerca Scientifica, 00133 Rome, Italy
2Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received 7 October 2013; Accepted 14 November 2013; Published 19 February 2014

Academic Editor: Vakhtang M. Kokilashvili

Copyright © 2014 Massimo A. Picardello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitable means. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.