arXiv subjects

Shaoqiang Shang

Publications and source records attributed to Shaoqiang Shang.

2 recordsLinked to original sources

arXiv2026

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

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 $G_δ$ subsets via sequences of dense open cones, this paper finally verifies that the product space satisfies the definition of a weak Asplund space.

math.FA↗

arXiv2026

A closed subspace of a Gateaux differentiability space is a Gateaux differentiability space : over 46 years of open problem solved

This paper establishes for the first time the iterative and rigid theory of weak$^{*}$ slices within a non-metric framework, demonstrating that dual convex sets under the pure weak$^{*}$ topology can achieve localization, diameter control, and fine structural analysis. It fundamentally transforms the traditional understanding of the geometric properties of weak$^{*}$ topology and thereby pioneers a new direction in non-metric weak$^{*}$ slice geometry. By developing a new technique involving intricate manipulations of weak$^{*}$ slices and a carefully designed iterative selection process, we prove that if $M$ is a closed subspace of a G$\mathrm{\hat{a}}$teaux differentiability space $X$, then $M$ is a G$\mathrm{\hat{a}}$teaux differentiability space. As a Corollary, we get that if $X$ is a weak Asplund space and $M$ is a closed subspace of $X$, then $X$ is a G$\mathrm{\hat{a}}$teaux differentiability space. Thus, we definitively solve an open problem raised 46 years ago by D.G. Larman and R.R. Phelps (J. London Math. Soc., 20(1979), 115--127).

math.FA↗