I was working on two exercises from the introductory chapter in a functional analysis reader, wherein the second is a followup to the first. I managed to solve the first question, though my solution is quite long and I won't quote it here. The same problem is discussed here, but here is the first problem as stated in my book:
Show that the $C^1$-norm $\|f\|_{C^1} := \|f\|_\infty + \|f'\|_\infty$ is indeed a norm on $C^1[0,1]$ and come up with a similar norm on $C^k[0,1]$ for all $k \in \mathbb{N}$. Show that $$ d(f,g) := \sum_{k=0}^\infty\frac{2^{-k}\|f^{(k)}-g^{(k)}\|_\infty}{1+\|f^{(k)}-g^{(k)}\|_\infty} $$ defines a metric on $C^\infty[0,1]$ which is also translation-invariant. Can you come up with a norm $\|..\|_{C^\infty}$ on $C^\infty[0,1]$ such that there are real numbers $m,M > 0$ such that $$ \forall f \in C^\infty : m\|f\|_{C^\infty} \leq d(f,0) \leq M\|f\|_{C^\infty} $$
Now, verifying that the given metric is indeed a metric, and translation-invariant at that, I have already done. The open question about $C^k[0,1]$ is easy enough, and the answer to the final open question is no; this is not possible, because such a norm would induce a metric equivalent to $d$, but at the same time there are examples of functions that are continuous with respect to $d$ but not with respect to this hypothetical norm - a contradiction. All in agreement with the linked post as well.
All that is fine, but the next exercise is tricky to me:
Check that the metric $d$ from the previous exercise is also a metric on $$ E = \left\{ f \in C^\infty(\mathbb{R}) : t \mapsto e^{\lambda t}f(t) \in C_0(\mathbb{R}) \text{ for all } \lambda \in \mathbb{R} \right\} $$ and that it is also translation-invariant.
Here $C_0(\mathbb{R})$ denotes the set of all continuous functions on $\mathbb{R}$ whose values approach zero as $|t| \to \infty$. Verifying that all properties of a metric hold is completely analogous to the previous exercise, as well as checking rotational invariance: however, for this reasoning I need to prove first that $\|f^{(k)}-g^{(k)}\|_\infty$ exists for all $f, g \in E$ and $k \in \mathbb{N}$. This is easy for $k=0$: setting $\lambda = 0$ we immediately find that $f,g \in E$ implies $f,g \in C_0(\mathbb{R})$, and $\|.\|_\infty$ is well-defined on $C_0(\mathbb{R})$ so we are done. What I am wondering is how to prove that this norm is also well-defined for all other terms in the series: if so, it's easy to show that $d$ is bounded by 1, and is thus well-defined as the series always converges.
Could anyone provide some insight?
