On densely isomorphic normed spaces

Abstract

In the first part of our note we prove that every Weakly Lindelöf 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 ℓ 2 (ω 1 ) (or more generally every WLD space of density ω 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 Todorcević (under Martin Maximum) of an uncountable biorthogonal system in every non-separable Banach space.

Smooth norms in dense subspaces of Banach spaces

“…Let (e λ ) λ∈S be the canonical basis of ℓ 1 (S). Then, every dense subspace of ℓ 1 (S) contains a subspace isomorphic to span{e λ } λ∈S (see [15,Theorem 2.3]). By using Theorem 3.6, every dense subspace of ℓ 1 (S) contains a further dense subspace which admits an analytic norm.…”