On a theorem about Mosco convergence in Hadamard spaces

Abstract

Let $(f^n),f$ be a sequence of proper closed convex functions defined on a Hadamard space. We show that the convergence of proximal mappings $J^n_{\lambda}x$ to $J_{\lambda}x$, under certain additional conditions, imply Mosco convergence of $f^n$ to $f$. This result is a converse to a theorem of Bacak about Mosco convergence in Hadamard spaces.