Do normed linear spaces have the properties of a normal hausdorff space?
I just sat an exam and I couldn't work out how to prove something initially, then I assumed that normed linear spaces are $T_4$, and the proof was easy. Are they? Please say yes :)
The proof for the record was showing that if we have some normed linear space $X$, and the unit ball on this space has a compact surface, prove that $X$ is not infinite dimensional. Which after the exam I realized was a consequence of Riesz's lemma. But without that I took an open covering and using my $T_4$ properties there were disjoint open balls covering every element of the compact surface, which would cause contradiction for my finite subcovering over infinite elements.