Let $E$ be a $\mathbb R$-Banach space, $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $\iota$ be a continuous embedding of $\Theta$ into $C^1(E,E)$.

I would like to show that, given a compact $K\subseteq E$, there is a $c\ge0$ with $$\sup_{x\in K}\left\|{\rm D}(\iota f)(x)\right\|_{\mathfrak L(E)}\le c\left\|f\right\|_{\Theta}\;\;\;\text{for all }f\in\Theta.\tag1$$

If I'm not missing something, $$C^1(K,E):=\left\{\left.g\right|_K: g\in C^1(U,E)\text{ for some open neighborhood }U\text{ of }K\right\}$$ equipped with $$\left\|g\right\|_{C^1(K,\:E)}:=\max\left(\sup_{x\in K}\left\|g(x)\right\|_E,\sup_{x\in K}\left\|{\rm D}g(x)\right\|_{\mathfrak L(E)}\right)\;\;\;\text{for }g\in C^1(K,E)$$ should be a $\mathbb R$-Banach space. If that's true, we may be able to show $$\Theta\ni f\mapsto\left.(\iota f)\right|_K\tag2$$ is a continuous embedding of $\Theta$ into $C^1(K,E)$, from which the desired claim would follow.

Can we show this?

allcompact subsets of $\mathbb{R}^d$, never mind when $E$ is something more exotic (it is true for all 'nice' compact subsets of $\mathbb{R}^d$ of course). $\endgroup$ – DCM Oct 9 at 18:20