Previous |  Up |  Next


Title: Local/global uniform approximation of real-valued continuous functions (English)
Author: Hager, Anthony W.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 52
Issue: 2
Year: 2011
Pages: 283-291
Summary lang: English
Category: math
Summary: For a Tychonoff space $X$, $C(X)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and $C^{*}(X)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of $C^{*}(X)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C(X)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent to: Every compact space has AP. It is shown here that the class of spaces with AP contains all Lindelöf spaces and is closed under formation of topological sums. Thus, any locally compact paracompact space has AP. A paracompact space failing AP is Roy's completely metrizable space $\Delta$. (English)
Keyword: real-valued function
Keyword: Stone-Weierstrass
Keyword: uniform approximation
Keyword: Lindelöf space
Keyword: locally in
MSC: 06F20
MSC: 26E99
MSC: 41A30
MSC: 46E05
MSC: 54C30
MSC: 54C35
MSC: 54D20
MSC: 54D35
idZBL: Zbl 1240.54062
idMR: MR2849050
Date available: 2011-05-17T08:41:52Z
Last updated: 2013-09-22
Stable URL:
Reference: [BH74] Blair R., Hager A.: Extension of zero-sets and real-valued functions.Math. Z. 136 (1974), 41–52. MR 0385793, 10.1007/BF01189255
Reference: [D95] Darnel M.: Theory of Lattice-ordered Groups.Marcel Dekker, New York, 1995. Zbl 0810.06016, MR 1304052
Reference: [E89] Engelking R.: General Topology.Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321
Reference: [FGL65] Fine N., Gillman L., Lambek J.: Rings of quotients of rings of functions.McGill Univ. Press, 1965; republished by Network RAAG, 2005. Zbl 0143.35704, MR 0200747
Reference: [GJ60] Gillman L., Jerison M.: Rings of Continuous Functions.Van Nostrand, Princeton, N.J.-Toronto-London-New York, 1960. Zbl 0327.46040, MR 0116199
Reference: [H69] Hager A.: On inverse-closed subalgebras of $C(X)$.Proc. London Math. Soc. 19 (1969), 233–257. Zbl 0169.54005, MR 0244948, 10.1112/plms/s3-19.2.233
Reference: [H76] Hager A.: A class of function algebras (and compactifications, and uniform spaces).Sympos. Math. 17 (1976), 11–23. Zbl 0353.46014, MR 0425891
Reference: [H$\infty$] Hager A.: *-maximum $l$ preparation.
Reference: [HM02] Hager A., Martinez J.: $C$-epic compactifications.Topology Appl. 117 (2002), 113–138. Zbl 0993.54024, MR 1875905, 10.1016/S0166-8641(00)00119-X
Reference: [HR77] Hager A., Robertson L.: Representing and ringifying a Riesz space.Sympos. Math. 21 (1977), 411–431. Zbl 0382.06018, MR 0482728
Reference: [HR78] Hager A., Robertson L.: Extremal units in an Archimedean Reisz space.Rend. Sem. Mat. Univ. Padova 59 (1978), 97–115. MR 0547081
Reference: [HJ61] Henriksen M., Johnson D.: On the structure of a class of lattice-ordered algebras.Fund. Math. 50 (1961), 73–94. MR 0133698
Reference: [H47] Hewitt E.: Certain generalizations of the Weierstrass Approximation Theorem.Duke Math. J. 14 (1947), 419–427. Zbl 0029.30302, MR 0021662
Reference: [N73] Nyikos P.: Prabir Roy's space $\Delta$ is not $\mathbb N$-compact.General Topology and Appl. 3 (1973), 197–210. MR 0324657, 10.1016/0016-660X(72)90012-8
Reference: [R68] Roy P.: Nonequality of dimensions for metric spaces.Trans. Amer. Math. Soc. 134 (1968), 117–132. Zbl 0181.26002, MR 0227960, 10.1090/S0002-9947-1968-0227960-2
Reference: [S87] Sola M.: Roy's space $\Delta$ and its $\mathbb N$-compactification.Thesis, Univ. of S. Carolina, 1987.
Reference: [S48] Stone M.: The generalized Weierstrass approximation theorem.Math. Mag. 21 (1948), 167–184. MR 0027121, 10.2307/3029750


Files Size Format View
CommentatMathUnivCarolRetro_52-2011-2_8.pdf 233.0Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo