2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73792A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic nature which implies an interesting dichotomy for subspaces of Banach spaces. Combined with a result of Komorowski and Tomczak-Jaegermann, this gives a positive answer to Banach's problem. We then generalize the Ramsey-theoretic result and deduce a further dichotomy for Banach spaces with an unconditional basis.37 pages, published versionFunctional Analysis46B20 (Primary) 03E02, 03E15, 05D10, 46B03 (Secondary)text