## Abstract and Applied Analysis

### Function Spaces with Bounded ${L}^{p}$ Means and Their Continuous Functionals

Massimo A. Picardello

#### Abstract

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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 609525, 26 pages.

Dates
First available in Project Euclid: 7 October 2014

https://projecteuclid.org/euclid.aaa/1412687022

Digital Object Identifier
doi:10.1155/2014/609525

Mathematical Reviews number (MathSciNet)
MR3173284

#### Citation

Picardello, Massimo A. Function Spaces with Bounded ${L}^{p}$ Means and Their Continuous Functionals. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 609525, 26 pages. doi:10.1155/2014/609525. https://projecteuclid.org/euclid.aaa/1412687022

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