Is the restriction map $C^1\ni f\mapsto\left.f\right|_K$ a continuous map?

Question

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?

Answer 1

I agree with Pietro Majer's comment that (1) follows from the continuity of the restrictions $C^1(E,E)\to C^1(K,E)$ -- whatever you mean by this space!

Concerning (2) (in the finite dimensional case $E=\mathbb R^d$): If you define $C^1(K,E)$ as the space of restrictions of $C^1$-functions on open supersets of $K$ then, in general, it isn't complete with respect to the norm $$ \|g\|_K=\sup\{|g(x)|: x\in K\} +\sup\{|Dg(x)|: x\in K\}. $$ This was suspected in the comment of DCM and it is indeed well-known since the work of Whitney.

There is a recent paper of Leonhard Frerick, Laurent Loosveldt and myself (Continuously differentiable functions on compact sets, arXiv:2003.09681) on various definitions of $C^1(K)$. Theorem 5.1 implies that the space of restrictions is complete with respect to the norm above if and only if $K$ has finitely many components which are Whitney regular, i.e., the geodesic distance is equivalent to the euclidean distance. For general $K$ (which is equal to the closure of its interior) one should endow the space of restrictions with the (ugly) quotient norm $$ \|g\|=\inf\{\|f\|_{\mathbb R^d}: f\in C^1(\mathbb R^d), f|_K=g\}. $$ Whitney gave a simpler description for this. For general $K$ one should rather consider the space of Whitney jets $(f|_K, Df|K)$ than just the functions because "the" derivative might not be uniquely defined by a function defined just on $K$.

Answer 2

As an addendum to Jochen Wengenroth's answer: If you are willing to restrict your choice of compact sets somewhat (to those whose interior is dense), then you might find the answers to your questions (for $E$ finite dimensional) in the recent preprint

• Helge Glockner, Smoothing operators for vector-valued functions and extension operators, arXiv:2006.00254.

Note that the restriction to finite-dimensional spaces here is necessary as on one hand there are no compact sets with nonempty interior in infinite-dimensional spaces and in addition, the differentiability discussed in the paper contains with the Fréchet differentiability you asked for on finite-dimensional spaces.