The product of a weak Asplund space and a one-dimensional space is a weak Asplund space: over 45 years of open problem solved

Abstract

In this paper, authors prove that if $X$ is a weak Asplund space, then the space $X\times R$ is a weak Asplund space. Thus the author definitely answered an open problem raised by D.G. Larman and R.R. Phelps for 45 years ago (J. London. Math. Soc. (2), 20(1979), 115--127). This paper conducts analysis combining Banach-Mazur game theory, properties of maximal monotone operators and the G$\mathrm{\hat{a}}$teaux differentiability of Minkowski functionals, and proposes the iterative perturbed Minkowski convex cone approximation game method. It establishes a theoretical framework for verifying the existence of dense differentiable sets of convex functions on product spaces. By using projection mappings to correlate the properties of convex functions on the original space and the product space, and constructing dense $G_{\delta}$ subsets via sequences of dense open cones, this paper finally verifies that the product space satisfies the definition of a weak Asplund space.