On the unique predual problem for Lipschitz spaces

Abstract

For any metric space X, the predual of Lip(X) is unique. A previous version of this manuscript, which is also the published version (Math. Prof. Cambridge Philos. Soc. 165 (2018), 467-473), additionally stated "If X has finite diameter or is complete and convex -- in particular, if it is a Banach space -- then the predual of Lip_0(X) is unique." However, the proof of a crucial lemma, Lemma 3.1 in the previous version, was faulty. The error in that proof lay in assuming that the limit, for the topology induced by W, of a net in the unit ball would have to lie in the unit ball. But we do not know that W is 1-norming. This error was pointed out by Manuel Gonzalez, as relayed to me by Ruben Medina. The reduction from "complete and convex" to "finite diameter" is still valid, and is retained in the present version.

6 pages