Let X be a complex Hausdorff topological vector space and let Y be a complex Hausdorff locally convex and quasi-complete topological vector space. Let P stand for a generalized polynomial map from X to Y of the form \(P=P_ 0+P_ 1+...+P_ N\), \(P_ N\neq 0\), where \(P_ m: X\to Y\), \(m=1,2,...,N\), are m-homogeneous generalized polynomial maps and \(P_ 0: X\to Y\) is a constant map and let P’(x,x) be the directional derivative of P at x in the direction of x.
Main result: Let \(\Omega\) and \(\Gamma\) be closed balanced convex sets in X and Y, respectively, and let p and q be their gauges. If \(P: X\to Y\) is a generalized polynomial map of degree N and \(P: \Omega\) \(\to \Gamma\), then, for each \(x\in \Omega\), the inequality \(q\circ P'(x,y)\leq Np(x)\) holds. The result is best possible and equality holds for an N- homogeneous generalized polynomial map P and \(x\in \Omega\), such that \(q\circ P(x)=p(x)=1\). (In case \(X=Y={\mathbb{C}}\) this result is due to Bernstein.)