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Abstract

This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_∞$ of the corresponding functions in $A(l_∞)$. These results are achieved by studying the homomorphisms on the function algebras in question, an idea that is also reversed in order to obtain new results for the set of homomorphisms on these algebras.