In introductory functional analysis one learns that every normed linear space with a Schauder basis is separable. The converse of this was a famous question raised by Banach- does every separable Banach space have a Schauder basis? Since almost all known separable Banach spaces had been shown to possess a Schauder basis it was believed that this must be true. But in 1972 Enflo constructed a counterexample to this.
For this achievement of his Enflo was awarded a live goose by Stanislaw Mazur.
See this.
