What is meant by saying $F:\mathbb R^2\to \mathbb R$ is infinitely differentiable?

Question

When we say $f:\mathbb R \to \mathbb R$ is infinitely differentiable at some $a\in \mathbb R$, then we mean the $n^{th}$ derivative $f^{(n)}(a)$ exists for all $n\in \mathbb N$. What is meant by saying $F:\mathbb R^2\to \mathbb R$ is infinitely differentiable?

We know, the total derivative of $F$ at some point $p=(a,b)\in \mathbb R^2$ is a particular bounded linear map $L_{p}:\mathbb R^2\to \mathbb R$; and the total derivative of $F$ is the map $D:\mathbb R^2\to L(\mathbb R^2,\mathbb R)$, defined as: $D(p)=L_p$, $p\in \mathbb R^2$, where $L(\mathbb R^2,\mathbb R)$ is the set of all bounded linear operator from $\mathbb R^2$ to $\mathbb R$. Now, when we say $F$ is two times differentiable at $(a,b)$, then do we mean that the map $D$ has a total derivative at $(a,b)$? And, proceeding this way, do we say $F$ is infinitely differentiable?

Answer 1

The following are all equivalent for a function $f:U\subset\Bbb{R}^n\to \Bbb{R}^m$ (where $U$ is open):

• For all $k\geq 0$, the iterated $k^{th}$ Frechet derivative, $D^kf$ (which maps $U$ into some big space of (multi-)linear maps, for example, when $k=2$, we have a map $D^2f:U\to L(\Bbb{R}^n;L(\Bbb{R}^n;\Bbb{R}^m))\cong L^2(\Bbb{R}^n;\Bbb{R}^m)$, i.e you can think of it as mapping into the space of bilinear maps $\Bbb{R}^n\times\Bbb{R}^n\to\Bbb{R}^m$), exists. This is the most a-priori meaningful definition for “infinitely differentiable on $U$”.
• For all $k\geq 0$, the iterated $k^{th}$ Frechet derivative, $D^kf$, exists and is continuous on $U$. This is the most a-priori meaningful definition for “infinitely continuously differentiable on $U$”.
• For all $k\geq 0$, and all $i\in\{1,\dots, m\}$, the iterated $k^{th}$ Frechet derivative of the $i^{th}$ component function, $D^k(f_i)$, exists.
• For all $k\geq 0$, and all $1\leq i\leq m$, the iterated $k^{th}$ Frechet derivative of the component function $f_i$,i.e $D^k(f_i)$, exists and is continuous on $U$.
• For all $j\in\{1,\dots, n\}$, the partial derivative $\frac{\partial f}{\partial x_j}:U\to\Bbb{R}^m$ exists and is continuous.
• For all $i\in\{1,\dots, m\}$ and all $j\in\{1,\dots, n\}$, the partial derivative $\frac{\partial f_i}{\partial x_j}:U\to\Bbb{R}$ exists and is continuous.

In practice, either of the last two bullet points are most easily verified. If $f$ satisfies any of these conditions, we write $f\in C^{\infty}(U,\Bbb{R}^m)$ (or in some cases, $f\in \mathcal{E}(U,\Bbb{R}^m)$).