I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides.
Let $\Omega \subset \mathbb{R}^N$ be an open set and
$C_K^\infty(\Omega) = \{ \phi \in C^\infty(\Omega) : \mathrm{supp}(\phi)\subset K \}$.
This space has has an infinite family of norms for each $p \in \mathbb{N}$, and compact $K \subset \Omega$ – the maximum of the partial derivative:
$$ ||\phi||_{p,K} = \max_{|\alpha|\leq p} \max_{x \in K} |\partial^\alpha \phi(x)|$$
Why is $C_c^\infty(\Omega)$ not a normed space, if we can compute a norm of a function in any compact subset?
Here "compact" means closed and bounded since we are dealing with $\mathbb{R}^n$.
Best Answer
I believe that something more precise is meant here, which is the following:
$C_c^{\infty}(\Omega)$ cannot be given a norm $\|~\|$ that will, on its own, generate the same topology as the family of seminorms $\{\|~\|_{p, K}\}$.
The intuition here is that if you had a finite family of good enough seminorms, you could add them to get one norm that generates the same topology. But in this case, the family being infinite, one cannot just add them all.
Of course, this is not a proof; to actually prove that this is not possible would involve finding some property that all normed spaces satisfy but this one doesn't (with this topology). I can't think of one right now.