Not really sure how to begin this one. Any help appreciated!
I have no idea what that means. I assume its the intersection of all the open balls of radius r given by the family of semi norms. There's a lot about this concept out there, especially with respect to locally convex spaces, but I still don't really understand it.
For the last part of the question, I think that you have to use a cardinality argument. You want to find a sequence in which converges to 0 (in the topology of pointwise convergence) so slowly that if you multiply its terms by any sequence of scalars that tends to infinity then the resulting sequence no longer converges to 0.
The set of all sequences of integers that tend to infinity has cardinality , which is the same as the cardinality of [0,1]. So there exists a bijective map . Define the function by .
Then for all , but
Finally, I leave you to check that if fails to converge to 0 for all sequences of integers tending to infinity, then the same will be true for all sequences of reals tending to infinity.