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).
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