In my previous Functional Analysis course where we studied Normed space,Metric space, Hilbert space Banach space etc, it was relatively easy for me to find counterexamples to certain theorems.
However, I am now taking an advanced Functional Analysis course and we are more focused on Topological Vector Space. I now find it harder to find counterexamples. For example, I was checking whether there are counterexamples to
- Every bounded set is compact?
- Every totally bounded set is compact?
- Every bounded set is totally bounded?
While I did find some counterexamples on this site, they are not in topological vector space. For example https://math.stackexchange.com/a/2086003/243942
So I was wondering whether I can use these counterexamples in topological vector space anyway?
Does a counterexample that work Normed space or Metric space or Hilbert space or Banach space also work in Topological vector space?
Yes, an example space that is normed, metric or Hilbert, Banach is a TVS as long as the linear space operations are continuous (automatic for a norm, but not necessarily for any metric space that is a vector space..).
The whole point of the theory of topological vector spaces is to generalise, or create a more general setting for results in the classic examples. Most functional analysis is done (I think) in Banach spaces and Hilbert spaces (completeness is nice to have).
So the unit ball in a an infinite-dimensional normed space is bounded (also in the TVS sense), but not compact. That still stands.
For total boundedness (so Q2) normed examples also exist, even in $\Bbb R^n$...
Etc.