English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

A well known result of H. Attouch states that the Mosco convergence of asequence of proper convex lower semicontinuous functions defined on a Hilbertspace is equivalent to the pointwise convergence of the associated Moreauenvelopes. In the present paper we generalize this result to Hadamard spaces.More precisely, while it has already been known that the Mosco convergence of asequence of convex lower semicontinuous functions on a Hadamard space impliesthe pointwise convergence of the corresponding Moreau envelopes, the converseimplication was an open question. We now fill this gap.  Our result has several consequences. It implies, for instance, theequivalence of the Mosco and Frol\'ik-Wijsman convergences of convex sets. Asanother application, we show that there exists a complete metric on the cone ofproper convex lower semicontinuous functions on a separable Hadamard space suchthat a sequence of functions converges in this metric if and only if itconverges in the sense of Mosco.

Convergence of Functions and their Moreau Envelopes on Hadamard Spaces