Open Access
2014 Function Spaces with Bounded L p Means and Their Continuous Functionals
Massimo A. Picardello
Abstr. Appl. Anal. 2014(SI25): 1-26 (2014). DOI: 10.1155/2014/609525


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 L p 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.


Download Citation

Massimo A. Picardello. "Function Spaces with Bounded L p Means and Their Continuous Functionals." Abstr. Appl. Anal. 2014 (SI25) 1 - 26, 2014.


Published: 2014
First available in Project Euclid: 7 October 2014

MathSciNet: MR3173284
Digital Object Identifier: 10.1155/2014/609525

Rights: Copyright © 2014 Hindawi

Vol.2014 • No. SI25 • 2014
Back to Top