On topological classification of normed spaces endowed with the weak topology or the topology of compact convergence

TL;DR

This paper investigates the topological classification of normed spaces with weak and compact convergence topologies, revealing conditions under which these spaces are homeomorphic, especially for spaces with separable duals and hyperplane isomorphisms.

Contribution

It establishes a topological classification framework for normed spaces with weak and compact convergence topologies, linking homeomorphism classes to properties of duals and hyperplane isomorphisms.

Findings

Spaces with separable duals are classified by their classes of homeomorphic bounded subsets.

For hyperplane-isomorphic Banach spaces with separable duals, weak and compact convergence topologies are homeomorphic.

Weak and compact convergence topologies are homeomorphic to products involving the weak unit ball and ^ infinity.

Abstract

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the dual space $X^*$), we consider the topology $c$ of uniform convergence on compact subsets of $X^*$. It is known that this topology coincides with the weak topology on bounded subsets of $X$, but unlike to the latter has much better topological properties (e.g., is stratifiable). We prove that for normed spaces $X,Y$ with separable duals the spaces $(X,weak)$, $(Y,weak)$ are sequentially homeomorphic if and only if $\mathcal W(X)=\mathcal W(Y)$, where $\mathcal W(X)$ is the class of topological spaces homeomorphic to closed bounded subsets of $(X,weak)$. Moreover, if $X,Y$ are Banach spaces which are isomorphic to their hyperplanes and have separale duals, then the spaces $(X,weak)$ and $(Y,weak)$ are sequentially homeomorphic if and only of the spaces $(X,c)$ and $(Y,c)$ are homeomorphic. To prove this result, we show that for a normed space $X$ which is isomorphic to its hyperpane and has separable dual, the space $(X,c)$ (resp. $(X,weak)$) is (sequentially) homeomorphic to the product $B\times\mathbb R^\infty$ of the weak unit ball $B$ of $X$ and the linear space $\mathbb R^\infty$ with countable Hamel basis and the strongest linear topology.