Let and be topological vector spaces over , and be continuous and additive, i.e.
Then is linear, i.e.
It follows that . Let . By additivity,
It follows that . Thus preserves negation. We claim that , for all . Since
the claim holds for . Assume the claim holds for , with . Then
Thus preserves multiplication by natural numbers. Let . Then if ,
Thus preserves multiplication by integers. Let , with . Then
Thus , and preserves multiplication by inverses of non-zero integers. Let . Then
Thus preserves multiplication by rational numbers. Let be a sequence of rational numbers converging to a real number , i.e. . Then
where the limit can be moved out of by continuity of . The can be moved out of the limit by the continuity of vector multiplication (which holds by definition for a topological vector space). Thus preserves multiplication by real numbers. Therefore, is both homogeneous and additive, and thus linear.