On densely isomorphic normed spaces

TL;DR

This paper investigates the structure of non-separable Banach spaces, showing they contain dense subspaces that are not densely isomorphic and exploring the existence of biorthogonal systems under certain set-theoretic assumptions.

Contribution

It proves the existence of dense subspaces not densely isomorphic in non-separable WLD Banach spaces and constructs dense subspaces with no uncountable biorthogonal systems under CH.

Findings

Non-separable WLD spaces contain non-densely isomorphic dense subspaces.

Under CH, there exists a dense subspace of () with no uncountable biorthogonal system.

Results bridge known constructions by Kunen and Todorvi
07.

Abstract

In the first part of our note we prove that every Weakly Lindel\"of Determined (WLD) (in particular, every reflexive) non-separable Banach $X$ space contains two dense linear subspaces $Y$ and $Z$ that are not densely isomorphic. This means that there are no further dense linear subspaces $Y_0$ and $Z_0$ of $Y$ and $Z$ which are linearly isomorphic. Our main result (Theorem B) concerns the existence of biorthogonal systems in normed spaces. In particular, we prove under the Continuum Hypothesis (CH) that there exists a dense linear subspace of $\ell_2(\omega_1)$ (or more generally every WLD space of density $\omega_1$) which contains no uncountable biorthogonal system. This result lies between two fundamental results concerning biorthogonal systems, namely the construction of Kunen (under CH) of a non-separable Banach space which contains no uncountable biorthogonal system, and the construction of Todor\u{c}evi\'c (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space.