Some Banach spaces added by a Cohen real

We study certain Banach spaces that are added in the extension by one Cohen real. Specifically, we show that adding just one Cohen real to any model adds a Banach space of density ℵ1ℵ1 which does not embed into any such space in the ground model (Theorem 1.1). Moreover, such a Banach space can be chosen to be UG (Theorem 1.6). This has consequences on the isomorphic universality number for Banach spaces of density ℵ1ℵ1, which is hence equal to ℵ2ℵ2 in the standard Cohen model and the same is true for UG spaces. Analogous universality results for Banach spaces are true for other cardinals, by a different proof (Theorem 2.10(1)).