Space of All Smooth Lipschitz Functions This is a followup question to this post.  Let $F$ be the collection of all Lipschitz functions from $\mathbb{R}^d$ to itself, which admit $k>0$ derivatives everywhere and such that the $k^{th}$ differential is itself Lipschitz.  Is this space a complete metric space, under the metric
$$
d(f,g) \triangleq \sum_{i=0}^k \|f^{(i)}(x)-g^{(i)}(x)\| + Lip(f^{(k)}-g^{(k)})?
$$
Here $Lip(h)$ is the minimal Lipschitz constant of a Lipschitz function $h$. 
 A: Note the following:


*

*If $f_n$ is a sequence of bounded continuous functions that are Cauchy wrt the supremum norm, then a bounded continuous function $f$ exists so that $\|f_n -f\|\to0$.

*If $f_n$ is as above and additionally differentiable with $df_n$ being bounded continuous and Cauchy wrt sup-norm, then $f$ is differentiable and $df=\lim_n df_n$.

*If $f:\Bbb R^d\to \Bbb R^d$ has bounded differential, then $f$ is Lipschitz (with Lipschitz constant $\|df\|$).

*If $f_n$ is a sequence like in 1. and additionally $f_n$ are Lipschitz with the Lipschitz constants being $≤$ a common bound, then $f$ is Lipschitz.


Now combine this stuff to see:
If you have a sequence of $k$-times differentiable functions $f_n$ so that all $k$-derivatives are bounded and your sequence is Cauchy wrt the $k$ semi-norms $\|d^i f_n\|$, $i\in\{0,...,k-1\}$, then a limit $f$ exists and is also $k$-times differentiable (apply 1. and 2. $k$-times). It follows that the first $k-1$ differentials are Lipschitz (from 3.).
If you also ask that $f_n$ is Cauchy wrt the semi-norm $\mathrm{Lip}(d^k f)$, then 4. tells you that the limit $f$ also has the property that $d^k$ is Lipschitz.
Now note that a sequence is Cauchy wrt a finite amount of semi-norms $\|\cdot\|_{i}$ iff it is Cauchy wrt $\sum_{i} \|\cdot\|_i$. Thus if you have a sequence that is Cauchy wrt your norm, then it admits a limit that also has the properties of being $k$-times differentiable with the $k$-th derivative being Lipschitz. This means your space is complete.
