Approximation theorems for uniformly continuous functions
Author:
Anthony W. Hager
Journal:
Trans. Amer. Math. Soc. 236 (1978), 263273
MSC:
Primary 41A65; Secondary 41A30
MathSciNet review:
0510848
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Abstract: Let X be a set, A a family of realvalued functions on X which contains the constants, the weak uniformity generated by A, and the collection of uniformly continuous functions to the real line R. The problem is how to construct from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely Aequiuniform, orderone subsets of is uniformly dense in . Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely Auniform and uniformly locallyA (resp., uniformly locally piecewiseA) is uniformly dense in . Further, for any A, a finitely Auniform function is just a composition for some and F uniformly continuous on the range of in . Thus, such compositions are dense in . For , the compositions with are dense (B denoting bounded functions). So, in a sense, to know it suffices to know A and subspaces of the spaces , and to know , A and the spaces suffice.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197805108483
PII:
S 00029947(1978)05108483
Keywords:
StoneWeierstrass,
uniformly continuous,
uniform approximation
Article copyright:
© Copyright 1978
American Mathematical Society
