Let and be topological vector spaces over the reals, and let such that is continuous and additive, i.e.
In this post I will show that is then also linear, i.e.
Since , it follows that . Now
from which it follows that . Thus preserves negation. We now claim that if , then The proof is by induction. Since , the claim holds when . Assume that the claim holds for , with . Then
Thus preserves multiplication by natural numbers. Let . Then if ,
Thus preserves multiplication by integers. Let , with . Now
Thus , and preserves multiplication by inverses of non-zero integers. Let . Then
Thus preserves multiplication by rational numbers. Finally, let with 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.