Function Spaces with Bounded Lp Means and Their Continuous Functionals
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/609525 |
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| Summary: | 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 Lp 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. |
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| ISSN: | 1085-3375 1687-0409 |