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Let C [ 0, 1 ] be the Banach algebra of real valued continuous functions on [ 0, 1 ], provided with the supremum norm. For f, g ∈ C [ 0, 1 ] and balls Bf , Bg with center f and g, respectively, it is not necessarily true that f · g is in the interior of Bf · Bg. In the present paper we characterize those pairs f, g where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way. AMS-classification: 46B20. 46E15; keywords: continuous functions, open sets, Banach algebra. This paper is dedicated to Susanne Dierolf who passed away far too early.

Walk the dog, or: products of open balls in the space of continuous functions