The spaces here should have a topology, or continuity is pointless.

So, consider them to be real topological vector spaces (which is consistent with the fact that we need to have a linear map here).

Let be

additive (i.e. ) and continuous.

To show is linear, we show that for , we have (%).

We easily see that , for (*). Also, we notice that by additivity , so . So (*) also holds for negative n.

For , we have . Now (*) grants us (%) for the rationals.

For , consider a sequence of rationals We have

. Now and , thus

. qed