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This paper shows that the product of a Gâteaux differentiability space and a separable Banach space is again a Gâteaux differentiability space. A Banach space is called a Gâteaux differentiability space if every convex continuous function on it is Gâteaux differentiable at the points of a dense set. For more information about this class and its relatives—weak Asplund and Asplund spaces—see [LP], [NP], [A], [P], [F], [DGZ], [G]. Relatively little is known about the stability of the class of the Gâteaux differentiability spaces. In the late eighties, the second named author showed that if X is a Gâteaux differentiability space, then so is the product of X and the real line; see [P, Proposition 6.5]. The aim of this note is to extend this fact to separable spaces. Theorem. Let X be a Gâteaux differentiability space and let Y be an arbitrary separable Banach space. Then X × Y is also a Gâteaux differentiability space. Proof. We shall elaborate Ioffe’s penalty method used in the proof of [P, Proposition 6.5], while incorporating some ideas from Klee [K]. Let BX and BY denote the closed unit balls in X and Y respectively. Let f : X × Y → R be a convex continuous function. Let Ω be any nonempty open set in X × Y . We have to find a point in Ω where f is Gâteaux differentiable. By a suitable shift of variables we can arrange things so that (0, 0) ∈ Ω. Since f is continuous at (0, 0), we can find ∆ > 0 so that Ω contains S = 2∆BX × 2∆BY and that f is bounded on this “square”. Let M > 0 be a constant such that |f(x, y)| < M whenever (x, y) ∈ 2∆BX × 2∆BY . Let {yi; i ∈ N} be a dense subset in ∆BY . Define g : 2∆BX → R as g(x) = sup { f ( x, ∑ 2αiyi ) −tan (∑ αi ) ;αi ∈ R, ∑ αi < π 2 } , x ∈ 2∆BX . Received by the editors March 16, 2000. 2000 Mathematics Subject Classification. Primary 46B20, 46G05, 26E15, 58C20.

The product of a Gâteaux differentiability space and a separable space is a Gâteaux differentiability space