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Proof. If E has its weak topology then, as is well known, it is a subspace of a product of copies of the reals. Therefore, by Lemma 2 of [4], every discrete subgroup of E is finitely generated. Suppose E does not have its weak topology. If F denotes the vector space spanned by the unit vectors {e,} in the Banach space 11 then, by Theorem 1.4 of [5], F is a continuous linear image of a subspace of E. Noting that the subgroup of F generated by the {e,} has the discrete topology, we see that E has a discrete subgroup which is not finitely generated. The proof is complete.

Remark. The above result is also true for complex vector spaces.