Ring derivations on algebras of continuous functions (Q1894895)

A ringderivation on a topological algebra \(A\) is a map \(D\) from \(A\) to \(A\) satisfying the conditions 1) \(D(a+b)=D(a) +D(b)\) and 2) \(D(ab)=aD(b)+D(a)b\). Let \(G\) be an open subset of the real line or the complex plane and let \(A\) be one of the algebras (real or complex) \(C(G)\), \(C^{\infty}(G)\) or \(H(G)\) (holomorphic functions on \(G\)) equipped with the topology of uniform convergence on compact subsets of \(G\). The author gives a simple proof of the automatic continuity of every ringderivation on \(A\), a proof which can be extended to a large class of topological algebras.