A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual

Antonio Acuaviva

A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual

Functional Analysis 2026-07-14 v1

Authors: Antonio Acuaviva

Abstract

We construct a separable Banach space $X$ with a Schauder basis such that $B_X$ is not a uniformly continuous retract of $B_{X^{**}}$ . Consequently, $X$ is not a uniformly continuous retract of $X^{**}$ and hence is not a Lipschitz retract of $X^{**}$ .

Keywords

separable Banach space Schauder basis Lipschitz retract uniformly continuous retract bidual unit ball

Cite

@article{arxiv.2607.12935,
  title  = {A Separable Banach Space with a Schauder Basis Which Is Not a Lipschitz Retract of Its Bidual},
  author = {Antonio Acuaviva},
  journal= {arXiv preprint arXiv:2607.12935},
  year   = {2026}
}