  The category $Ban$ of Banach spaces and linear maps of norm $\leq 1$ is
locally $\aleph_1$-presentable but not locally finitely presentable. We prove,
however, that $Ban$ is locally finitely presentable in the enriched sense over
complete metric spaces. Moreover, in this sense, pure morphisms are just ideals
of Banach spaces. We characterize classes of Banach spaces approximately
injective to sets of morphisms having finite-dimensional domains and separable
codomains.

Category Theory;Functional Analysis