On the Weak* Convergence of Gâteaux Derivatives of Upper Envelopes

For convex functions on Banach space endowed with a uniformly Gâteaux differentiable norm two observations are presented: first, the Moreau envelope of a proper lower semicontinuous convex function is Gâteaux differentiable; second, if the Moreau envelopes of a sequence of lower semicontinuous convex functions $\{f_n\}_{n=1}^{\infty}$ create a pointwise convergent sequence, then Gâteaux derivatives of the envelopes (computed at a given point) are weakly$^*$ convergent.