Question: Consider $C^1(\mathbb{R})$ of bounded continuously differentiable function with bounded derivative, equipped with the norm $\|f\|_{C^1}\equiv \sup_{\mathbb{R}}(|f|)+\sup_{\mathbb{R}}(|f'|)$. While it is not a Banach space, I am wondering if we can "make it" a Banach space by intersecting it with $H^1(\mathbb{R})$. My question thus is:
Is the space $$ C^1(\mathbb{R})\cap H^1(\mathbb{R}) $$ Banach when equipped with the norm being $$ \max\left(\|f\|_{C^1},\|f\|_{H^1}\right)? $$
Context I am solving a PDE that was shown to be well-posed on a space very similar to the one above and it would really be useful if it could be a Banach space.
Reasoning I am thinking that by making the functions go to zero fast enough at $\pm \infty$, it might make a proof of completeness work out. I understand the proofs for $C^1([a,b])$ rely on the application of uniform convergence on $[a,b]$, which I am not sure how to apply when the interval is $\mathbb{R}$. Perhaps there is a way to make $C^1(\mathbb{R})$ Banach if we simply ask the functions in it converge fast anough to zero at $\pm\infty$?