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

Abstract

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).