Cone lattices of upper semicontinuous functions

Beer, Gerald

doi:10.1090/S0002-9939-1982-0663871-9

Cone lattices of upper semicontinuous functions
HTML articles powered by AMS MathViewer

by Gerald Beer PDF

Proc. Amer. Math. Soc. 86 (1982), 81-84 Request permission

Abstract:

Let $X$ be a compact metric space. A well-known theorem of M. H. Stone states that if $\Omega$ is a vector lattice of continuous functions on $X$ that separates points and contains a nonzero constant function, then the uniform closure of $\Omega$ is $C(X)$. In this article we generalize Stone’s sufficient conditions to the upper semicontinuous functions on $X$ topologized in a natural way.

References

Gerald Beer, A natural topology for upper semicontinuous functions and a Baire category dual for convergence in measure, Pacific J. Math. 96 (1981), no. 2, 251–263. MR 637972
Gerald Beer, Upper semicontinuous functions and the Stone approximation theorem, J. Approx. Theory 34 (1982), no. 1, 1–11. MR 647707, DOI 10.1016/0021-9045(82)90110-1
Ennio De Giorgi and Tullio Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975), no. 6, 842–850 (Italian). MR 448194
Szymon Dolecki, Gabriella Salinetti, and Roger J.-B. Wets, Convergence of functions: equi-semicontinuity, Trans. Amer. Math. Soc. 276 (1983), no. 1, 409–430. MR 684518, DOI 10.1090/S0002-9947-1983-0684518-7

A generalized Weierstrass approximation theorem

R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8