Return to Answer

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 https://arxiv.org/abs/2003.09681 of Leonhard Frerick, Laurent Loosveldt and myself 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 $K$.

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 $K$.

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.

The is a recent paper https://arxiv.org/abs/2003.09681 of Leonhard Frerick, Laurent Loosveldt and myself 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 $K$.

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 https://arxiv.org/abs/2003.09681 of Leonhard Frerick, Laurent Loosveldt and myself 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 $K$.

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.

The is a recent paper https://arxiv.org/abs/2003.09681 of Leonhard Frerick, Laurent Loosveldt and myself 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 $K$.

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.

The is a recent paper https://arxiv.org/abs/2003.09681 of Leonhard Frerick, Laurent Loosveldt and myself 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 $K$.