Local compactness of $C^k(M,N)$ strong space.

Question

Let $M$ and $N$ be two $C^k$ manifolds with $k\geq 1$, with $M$ non compact. I know that $C^k(M,N)$ with its strong (Whitney) topology isn't metrizable and that it's a Baire space. Can I prove that it's a locally compact and/or Hausdorff space or do you have any counterexample ? Thank you!

Answer 1

The space is Hausdorff because if $f$ and $g$ differ at some point, they can be separated by $C^0$ neighborhoods which are larger than $C^k$ neighborhoods.

But not locally compact, not even sequentially. The issue is that the uniform bound on the $k$-th derivative of $f_n$ does not imply the existence of the $k$-th derivative at the limit. Consider the example $f_n(x)=(x^2+1/n)^{1/2}x^{k-1}$.