The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E$ be a real inner product space, with its norm $|\cdot|$ and distance $d$ as fixed there, and let $\mathrm{Sym}(E)$ be the set of bounded symmetric bilinear forms on $E$, with the metric $d_{\mathrm{Sym}}$ fixed there. Let $U\subseteq E$ be open in $(E,d)$ and let $u:U\to\mathbb{R}$. That $u$ is differentiable on $U$, and its gradient map $Du:U\to E$, are as defined there; that $u$ has a second derivative at a point of $U$, and its Hessian map $D^{2}u:U\to\mathrm{Sym}(E)$, are as defined there.

1. (Class $C^{1}$)(i) The function $u$ is of class $C^{1}$ on $U$ if it is differentiable on $U$ and its gradient map is continuous on $U$ as a map into $(E,d)$. We write $C^{1}(U)$ for the set of all such functions.

2. (Class $C^{2}$)(ii) The function $u$ is of class $C^{2}$ on $U$ if it is of class $C^{1}$ on $U$, has a second derivative at every point of $U$, and its Hessian map is continuous on $U$ as a map into $(\mathrm{Sym}(E),d_{\mathrm{Sym}})$. We write $C^{2}(U)$ for the set of all such functions. The second condition is meaningful for a function differentiable on $U$: since $U$ is open, every $x\in U$ admits a positive $\rho\in\mathbb{R}$ with the open ball $B_{d}(x,\rho)$ contained in $U$, and $u$ is then differentiable at every point of $B_{d}(x,\rho)$, as The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space (i) requires.