Strengthened Stone-Weierstrass type theorem
The aim of the paper is to prove that if \(L\) is a linear subspace of the space \(\mathcal{C}(K)\) of all real-valued continuous functions defined on a nonempty compact Hausdorff space \(K\) such that \(\min(|f|, 1) \in L\) whenever \(f \in L\), then for any nonzero \(g \in \overline{L}\) (where...
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doaj-dfcd59cc57eb4e9f903c82373d2e873c2020-11-24T21:39:16ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742011-01-01314645650http://dx.doi.org/10.7494/OpMath.2011.31.4.6453143Strengthened Stone-Weierstrass type theoremPiotr Niemiec0Jagiellonian University, Institute of Mathematics, ul. Łojasiewicza 6, 30-348 Krakow, PolandThe aim of the paper is to prove that if \(L\) is a linear subspace of the space \(\mathcal{C}(K)\) of all real-valued continuous functions defined on a nonempty compact Hausdorff space \(K\) such that \(\min(|f|, 1) \in L\) whenever \(f \in L\), then for any nonzero \(g \in \overline{L}\) (where \(\overline{L}\) denotes the uniform closure of \(L\) in \(\mathcal{C}(K)\)) and for any sequence \((b_n)_{n=1}^{\infty}\) of positive numbers satisfying the relation \(\sum_{n=1}^{\infty} b_n = \|g\|\) there exists a sequence \((f_n)_{n=1}^{\infty}\) of elements of \(L\) such that \(\|f_n \|= b_n\) for each \(n \geq 1\), \(g = \sum _{n=1}^{\infty} f_n \) and \(|g|= \sum _{n=1}^{\infty} |f_n| \). Also the formula for \(\overline{L}\) is given.http://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3143.pdfStone-Weierstrass theoremfunction lattices |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Piotr Niemiec |
spellingShingle |
Piotr Niemiec Strengthened Stone-Weierstrass type theorem Opuscula Mathematica Stone-Weierstrass theorem function lattices |
author_facet |
Piotr Niemiec |
author_sort |
Piotr Niemiec |
title |
Strengthened Stone-Weierstrass type theorem |
title_short |
Strengthened Stone-Weierstrass type theorem |
title_full |
Strengthened Stone-Weierstrass type theorem |
title_fullStr |
Strengthened Stone-Weierstrass type theorem |
title_full_unstemmed |
Strengthened Stone-Weierstrass type theorem |
title_sort |
strengthened stone-weierstrass type theorem |
publisher |
AGH Univeristy of Science and Technology Press |
series |
Opuscula Mathematica |
issn |
1232-9274 |
publishDate |
2011-01-01 |
description |
The aim of the paper is to prove that if \(L\) is a linear subspace of the space \(\mathcal{C}(K)\) of all real-valued continuous functions defined on a nonempty compact Hausdorff space \(K\) such that \(\min(|f|, 1) \in L\) whenever \(f \in L\), then for any nonzero \(g \in \overline{L}\) (where \(\overline{L}\) denotes the uniform closure of \(L\) in \(\mathcal{C}(K)\)) and for any sequence \((b_n)_{n=1}^{\infty}\) of positive numbers satisfying the relation \(\sum_{n=1}^{\infty} b_n = \|g\|\) there exists a sequence \((f_n)_{n=1}^{\infty}\) of elements of \(L\) such that \(\|f_n \|= b_n\) for each \(n \geq 1\), \(g = \sum _{n=1}^{\infty} f_n \) and \(|g|= \sum _{n=1}^{\infty} |f_n| \). Also the formula for \(\overline{L}\) is given. |
topic |
Stone-Weierstrass theorem function lattices |
url |
http://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3143.pdf |
work_keys_str_mv |
AT piotrniemiec strengthenedstoneweierstrasstypetheorem |
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1725931677393354752 |