I try to read into the theory of distributions and there is one thing which bothers me. I read that a distribution is a linear, continuous functional from the space of test functions, which, depending on the author, is sometimes defined as the Schwartz space and sometimes as $C^\infty_K$ (smooth functions with compact support).
For starters, where does this ambiguity come from and does it matter?
Now for the case $C^\infty_K$, I found two definitions of the associated topology. One as the final topology induced by the inclusion maps from $C^\infty(K)$ (I already understand how the topology on those spaces is defined) and one as the final locally convex topology induced by those maps, i.e. the finest locally convex topology which makes those maps continuous.
I couldn't proof their equivalence, i.e. how do I show that the final topology induced by those maps is already linear and locally convex?