## Approximation theorems for uniformly continuous functions

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- by Anthony W. Hager PDF
- Trans. Amer. Math. Soc.
**236**(1978), 263-273 Request permission

## Abstract:

Let*X*be a set,

*A*a family of real-valued functions on

*X*which contains the constants, ${\mu _A}$ the weak uniformity generated by

*A*, and $U({\mu _A}X)$ the collection of uniformly continuous functions to the real line

*R*. The problem is how to construct $U({\mu _A}X)$ from

*A*. The main result here is:

*For A a vector lattice, the collection of suprema of countable, finitely A-equiuniform, order-one subsets of*${A^ + }$

*is uniformly dense in*$U({\mu _A}X)$. Two less technical corollaries:

*If A is a vector lattice*(

*resp., vector space*),

*then the collection of functions which are finitely A-uniform and uniformly locally-A*(

*resp., uniformly locally piecewise-A*)

*is uniformly dense in*$U({\mu _A}X)$.

*Further, for any A, a finitely A-uniform function is just a composition*$F \circ ({a_1}, \ldots ,{a_p})$

*for some*${a_1}, \ldots ,{a_p} \in A$

*and F uniformly continuous on the range of*$({a_1}, \ldots ,{a_p})$

*in*${R^p}$. Thus, such compositions are dense in $U({\mu _A}X)$. For $BU({\mu _A}X)$, the compositions with $F \in BU({R^p})$ are dense (

*B*denoting bounded functions). So, in a sense, to know $U({\mu _A}X)$ it suffices to know

*A*and subspaces of the spaces ${R^p}$, and to know $BU({\mu _A}X)$,

*A*and the spaces ${R^p}$ suffice.

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## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**236**(1978), 263-273 - MSC: Primary 41A65; Secondary 41A30
- DOI: https://doi.org/10.1090/S0002-9947-1978-0510848-3
- MathSciNet review: 0510848