For a positive continuous weight function ν on an open subset G of CN, let Hv(G) and Hv0(G) denote the Banach spaces (under the weighted supremum norm) of all holomorphic functions f on G such that ν f is bounded and ν f vanishes at infinity, respectively. We address the biduality problem as to when Hν(G) is naturally isometrically isomorphic to Hν0(G)**, and show in particular that this is the case whenever the closed unit ball in Hν0(G) in compact-open dense in the closed unit ball of Hν(G).
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