A simple proof of the sum formula (Q2721600)

Let \(X\) be a real Banach space with dual \(X^*\). If \(f,g: X\to \mathbb R\cup\{+\infty\}\) are two proper, convex, lower semi-continuous functions then \(\partial f(x) + \partial g(x) \subseteq \partial (f+g)(x),\) where \(\partial h(x) \subset X^*\) stands for the subdifferential of a convex function \(h:X\to \mathbb R\cup\{+\infty\}\). The equality sign in the above inclusion holds only under some supplementary conditions on the functions \(f,g\). One such condition was given by \textit{H. Attouch} and \textit{H. Brezis}, in ``Aspects of mathematics and its Applications'', Collect. Pap. Hon. L. Nachbin, (J. A. Barroso, Editor), pp. 125-133, North Holland (1986; Zbl 0642.46004), namely: \(\text{cl-lin}(\operatorname {dom} f- \operatorname {dom} g) = X\). The authors give a new, simple, short proof of this result. They uses some ideas of \textit{S. Simons} [Trans. Am. Math. Soc. 350, No. 7, 2953-2972 (1998; Zbl 0901.47034)], who gave another proof of this result.