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Abstract

In this expository article, the authors investigate bounding sets and evaluating properties of homomorphisms for algebras A(E) of functions on a real Banach space E. A subset B E is said to be A-bounding if for all f 2 A(E), supx2B |f(x)| < 1. A(E) is said to be single-set evaluating if for every homomorphism 2 Hom A(E) and every f 2 A(E), (f) 2 f(E), and it is said to be sequentially evaluating if for every homomorphism and every sequence (fn) A(E), there is a 2 E such that (fn) = fn(a) for every n. The following are among the results described: If A(E) contains the algebra of rational functions on E, then each A-bounding set is relatively compact in E provided that there is a function f 2 A(`1) that is unbounded on the unit vectors of `1. As a consequence, for every E the C1-bounding sets are relatively compact. Let AE(E) denote the set of functions f : E ! R which are a pointwise (infinite) sum of polynomials, and let RAE(E) denote the smallest inverse-closed algebra containing AE(E). If E is weakly Lindel¨of with the Dunford-Pettis property and E does not contain a copy of `1, then E = Hom(AE(E)) = HomRAE(E).