02.03.2012

Let and be topological vector spaces over , and be continuous and additive, i.e.

Then is linear, i.e.

By additivity,

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.

QED.