Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space, with its inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$ as fixed there. Let $\mathrm{Sym}(H)$ be the set of bounded symmetric bilinear forms on $H$, with the identity form $I=I_{H}$, its multiples $cI$ and the zero form $0_{\mathrm{Sym}}$, as fixed there. For $b\in\mathrm{Sym}(H)$, the operator represented by $b$ is written $T_{b}$; it belongs to the set $\mathcal{L}(H)$ of bounded linear maps from $H$ to itself. The classes $C^{1}(W)$ and $C^{2}(W)$ of a subset $W\subseteq H$ open in $(H,d)$, together with the gradient $Du(x)$ and the Hessian $D^{2}u(x)$, are as defined there; the set $H$ is itself open in $(H,d)$, since every open ball of $(H,d)$ is a subset of $H$.

Then the following hold.

1. (Affine functions)(i) Let $p\in H$ and $c\in\mathbb{R}$, and let $a:H\to\mathbb{R}$ be given by

$$a(x)=\langle p,x\rangle+c .$$

Then $a\in C^{2}(H)$, and $Da(x)=p$ and $D^{2}a(x)=0_{\mathrm{Sym}}$ for every $x\in H$.

2. (Quadratic forms)(ii) Let $b\in\mathrm{Sym}(H)$ and let $Q:H\to\mathbb{R}$ be given by

$$Q(x)=\tfrac{1}{2}\,b(x,x).$$

Then $Q\in C^{2}(H)$, and $DQ(x)=T_{b}x$ and $D^{2}Q(x)=b$ for every $x\in H$.

3. (Multiples of the squared distance to a point)(iii) Let $\alpha\in\mathbb{R}$ and $y_{0}\in H$, and let $q:H\to\mathbb{R}$ be given by

$$q(x)=\tfrac{\alpha}{2}\,|x-y_{0}|^{2}.$$

Then $q\in C^{2}(H)$, and $Dq(x)=\alpha\,(x-y_{0})$ and $D^{2}q(x)=\alpha I$ for every $x\in H$.

4. (Restriction to an open subset)(iv) Let $W\subseteq H$ be open in $(H,d)$. Then the restrictions to $W$ of the functions $a$, $Q$ and $q$ of claims 1, 2 and 3 belong to $C^{2}(W)$, and their gradients and Hessians at every $x\in W$ are those given in those claims.