For a positive continuous weight function on an open subset of , let and denote the Banach spaces, under the norm , of holomorphic functions on such that is bounded and vanishes at infinity on , respectively. The principal question addressed in the paper under review is whether is (isometrically isomorphic to) the bidual of . This problem may be viewed as an analogue to weighted function spaces because of the fact that is the bidual of .
An affirmative result (that is the bidual of was obtained in 1970 by Rubel and Shields for the case where is the open unit disc in and is a radial weight function vanishing at the boundary; that is, for all and . This result, cited by the authors here, occurs as a special case of the main result of this paper. Specifically, Theorem 1.1 shows that is always (isometrically isomorphic to) the dual space of a Banach space and that is the bidual of if and only if the unit ball of is dense in the unit ball of in the compact-open topology. This density criterion is then used to obtain the Rubel-Shields result as well as other more general examples. Theorem 2.3 shows that, if the set is balanced and if is radial, then is the bidual of whenever the latter space contains all polynomials.
The final section of the paper concerns the predual of and the structure of when equipped with various topologies, most notably the bounded weak star topology.