An abstract υ-metric was introduced by Ball and Sasane, with a view towards extending the classical υ-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an additional concrete special instance of the abstract υ-metric, by verifying all the assumptions demanded in the abstract set-up. This example links the abstract υ-metric with the one proposed by Vinnicombe as a candidate for the υ-metric for nonrational plants.