If $K$ is a topological field, does the forgetful functor from the category of topological $K$-vector spaces to the category of $K$-vector spaces preserve all limits and colimits?
This is motivated by me needing to calculate some limits and colimits of topological vector spaces, and it would be immensely helpful if I at least knew what the underlying vector space was. But somehow my google search hasn't returned anything useful.
I know that in the category of topological spaces, this is true since the forgetful functor has a left and right adjoint. I also know that for topological groups, this is true and the resulting group is just the underlying group equipped with the respective initial / final topology. This suggestes that it should be true also in this case.
