This paper studies the coincidence of the T fjj and T$ topologies on the space of holomorphic functions defined on an open subset U of a Banach space. Dineen and Mujica proved that T fjj = T$ when U is a balanced open subset of a separable Banach space with the bounded approximation property. Here, we study the T W = Tg problem for several types of non-balanced domains U.
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