Approximability of separable Banach spaces by a common vanishing gauge

Determine whether every separable Banach space is approximable in the following sense: whether there exists a vanishing gauge omega such that, for every compact set K and every varepsilon>0, a uniformly continuous map phi can be found with relatively compact image, sup_{xin K}\|x-phi(x)\|<varepsilon, and global modulus omega_phileqomega.

Background

The paper studies approximation of Banach-space operators and identity maps by finite-dimensional-range maps whose stability is controlled uniformly by a common modulus. After proving an equivalence between Lipschitz decompositions and the Lipschitz bounded approximation property, it turns to Kalton's weaker framework based on vanishing gauges and almost-Lipschitz bounds.

A Banach space is defined to be omega-approximable if, for some vanishing gauge omega, every compact subset can be approximated arbitrarily accurately by a uniformly continuous map with relatively compact image and modulus bounded by omega. The authors explicitly leave unresolved whether every separable Banach space has this property for some vanishing gauge.