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

Antonio Acuaviva

functional analysis

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

arXiv:2607.12935

summary

The paper constructs a separable Banach space with a Schauder basis whose unit ball is not a uniformly continuous (and thus not Lipschitz) retract of its bidual’s unit ball, showing the space itself is not a Lipschitz retract of its bidual.

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 $X^{**}$ .

Topics & keywords

#banach spaces #schauder basis #lipschitz retract #bidual #uniform continuityseparable Banach spaceSchauder basisLipschitz retractbidualuniformly continuous retractunit ball