Let X be a compact
Hausdorff space and denote the space of all real valued continuous functions on X by
C(X,R). With pointwise operations this space becomes a linear space which we
norm by defining ∥f∥ = sup_{x∈X}f(x) whenever f ∈ C(X,R). A subset ℱ of
C(X,R) is said to be point separating if for any two distinct points x and y in X
there is an f in ℱ with f(x)≠f(y). Let Z[F] denote the ring of all polynomials in
elements of ℱ which have integral coefficients. Thus an element q of Z[ℱ] is an
element of C(X,R) with the special form
where the a’s are integers, the f’s belong to ℱ and the r’s are nonnegative integers.
Such q are our integral polynomials. If X is a subset of ndimensional Euclidean
space and ℱ is taken to be the set of n coordinate projections, then the elements of
Z[ℱ] are polynomials in the usual sense.
