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

2025/10/29 by Shaoqiang Shang, Shang, Shaoqiang
Computer Science · Mathematics · #Advanced Banach Space Theory #Fixed Point Theorems Analysis #Optimization and Variational Analysis #math.FA

paper · pdf · doi:10.48550/arxiv.2510.26827

arxiv created 2026/08/01 · arxiv updated 2026/08/04

Abstract

In this paper, authors prove that if X is a weak Asplund space, then the space X× 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\mathrmateaux 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_δ subsets via sequences of dense open cones, this paper finally verifies that the product space satisfies the definition of a weak Asplund space.

Related

• On the product of Weak Asplund locally convex spaces
• Asplund spaces and the finest locally convex topology
• Banach spaces where convex combinations of relatively weakly open\n subsets of the unit ball are relatively weakly open
• Note on Kadets Klee property and Asplund spaces
• On the Ascoli property for locally convex spaces and topological groups