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

TL;DR

This paper proves that the product of a weak Asplund space with a one-dimensional space remains weak Asplund, solving a 45-year-old open problem and advancing the understanding of stability in Banach space theory.

Contribution

It establishes that the product of a weak Asplund space with a real line is weak Asplund, providing a significant theoretical result in Banach space stability.

Findings

Confirmed the product of a weak Asplund space and R is weak Asplund.

Developed a framework using Banach-Mazur game theory and monotone operators.

Enhanced stability theory of weak Asplund spaces with applications in optimization and PDEs.

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). The study constructs a framework for proving the existence of densely differentiable sets of convex functions in product spaces through the analysis of Banach-Mazur game theory, maximal monotone operator properties, and the Gateaux differentiability of Minkowski functionals. By associating the convex function properties of the original space and product space via projection mappings, and utilizing sequences of dense open cones to construct $G_{\delta}$-dense subsets, the research ultimately demonstrates that the product space is a weak Asplund space. This work not only enriches the stability theory of weak Asplund spaces and their products with one-dimensional spaces but also provides crucial theoretical support for applications in convex optimization, weak solution construction for partial differential equations, and stochastic analysis.