The Banach space of bounded Dirichlet finite harmonic functions on an open Riemann surface will be seen to be reflexive and also separable if and only if the underlying Riemann surface does not carry any unbounded Dirichlet finite harmonic function.
"Banach spaces of bounded Dirichlet finite harmonic functions on Riemann surfaces." Kodai Math. J. 36 (1) 15 - 37, March 2013. https://doi.org/10.2996/kmj/1364562715