By Jorge Alberto Barroso

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**Example text**

As is well known, an entire function f(z) is said to be of exponential type if there exist numbers M > 0 and T > 0 such that If(z)1 ,,;;;; Me 1"lzl for all z E C. The greatest lower bound of the numbers T for which this inequality holds is called the exponential type of f. Nachbin's generalization consists in replacing the exponential function in the above definition by a function '/t(t) = :L nEN '/tnt n, where 1/'n > 0 and '/tn+J/V'n tends decreasingly to 0 as n ~ 00. The ratio test for convergence shows that '/t is an entire function.

Bishop considers a subalgebra d of cg(K) and calls a set M c K antisymmetric with respect to d if I(x) == I(y) for all x, y E M and for all real-valued functions in d. If d is self-adjoint, in particular if the functions in d are real-valued, then M is antisymmetric if and only if all functions in d are constant on M. g. the closed unit disk clos B1(O) in C and let d(B1(O» be the subalgebra of cg(clos B 1(O), C) consisting of all functions which are continuous on clos B 1(O) and holomorphic on the interior B 1(O) == {( E C] 1(1 < I} of clos B 1(O).

Deutsch to a proof of the Stone-Weierstrass theorem {ll}, {I2}. In the note [19] Leopoldo Nachbin establishes a Stone-Weierstrass theorem for the approximation of r-times continuously differentiable functions. He states it for n-dimensional differentiable manifolds but for the t ;,: J. Horvath / Life and Works of L. Nachbin 28 sake of simplicity I will state it for an open subset fl of the n-dimensional Euclidean space R", from where the general result follows anyway. Let r ~ 1 be an integer and denote by f{f'(fl) the algebra of all real-valued functions f defined on fl which have continuous partial derivatives s"] of orders lal = a1 + ...