# Prove that $E = \left\{ f \in C^\infty\left(\overline{\Omega}\right) : \lVert f \rVert_c < \infty \right\}$ is a Banach space

I am trying to determine whether a space is complete. I'll begin by setting up some notation. Let $$\Omega \subseteq \mathbb{R}^N$$ be an open set. For each $$n\in \mathbb{N}$$, define $$C^n\left(\overline{\Omega}\right)$$ be the space of $$n$$-times continuously differentiable functions endowed with the norm $$\lVert f \rVert_{C^n} = \sum_{\lvert \alpha \rvert \leq n} \sup_{x\in \overline{\Omega}}\lvert \partial^\alpha f(x)\rvert.$$ Given a sequence $$(c_n)$$ of positive numbers, set $$\lVert f \rVert_c = \sum_{n = 1}^\infty c_n \lVert f\rVert_{C^n}.$$ I wish to show that the normed space $$E = \left\{ f \in C^\infty\left(\overline{\Omega}\right) : \lVert f \rVert_c < \infty \right\}$$ is a Banach space.

To show this, I can begin by considering a Cauchy sequence $$(f_\alpha)$$ in $$E$$. One can easily show that there exists a function $$f\in C^\infty(\overline{\Omega})$$ such that $$f_\alpha \to f$$ in $$C^n$$ for every $$n\in \mathbb{N}$$. If I can also show that $$f_\alpha \to f$$ with respect to $$\lVert \cdot \rVert_c$$ then the proof is complete. In order to prove this, I would like to interchange between the infinite sum and the limit. The dominated convergence theorem does not seem to be sufficient to establish this.

Am I missing something? Is there some kind of condition that I could impose on the sequence $$(c_n)$$ so that $$E$$ becomes a Banach space or is this a hopeless endeavour. If $$E$$ is not a Banach space, is there a way I could go about proving that?

• What do you mean by $C^n(\overline{\Omega})$? Sep 15, 2020 at 10:52

## 1 Answer

It is a Banach space. Proof is by repeated application of the following one dimensional result:

Suppose $$f_n$$ is continuously differentiable for each $$n$$, $$f_n \to f$$ uniformly and $$f_n' \to g$$ uniformly. Then $$f$$ is differentiable and $$g=f'$$ ( so $$f_n' \to f'$$ uniformly).

• I know that $f_\alpha \to f$ with respect to the supremum norm (of every derivative). How does it follow that $f_\alpha \to f$ with respect to $\lVert f \rVert_c$? Sep 15, 2020 at 5:31
• Do you know how to show that the sequence space $\ell^{1}$ is complete? The proof here is similar. Unfortunately a detailed proof is too lengthy. Sep 15, 2020 at 5:35
• Oh of course! I think I made this problem harder than it needed to be. Thank you so much :-) Sep 15, 2020 at 6:11