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Countable projective limits of countable inductive limits, called PLB-spaces,of weighted Banach spaces of continuous functions have recently beeninvestigated by Agethen, Bierstedt and Bonet. In a previous article, the authorextended their investigation to the case of holomorphic functions andcharacterized when spaces over the unit disc w.r.t. weights whose decay,roughly speaking, is neither faster nor slower than that of a polynomial areultrabornological or barrelled. In this note, we prove a similarcharacterization for the case of weights which tend to zero logarithmically.

PLB-spaces of holomorphic functions with logarithmic growth conditions