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

Throughout, $H$ is open in $(H,d)$, as noted in the statement, and we use that $0\le\lVert b\rVert$ for $b\in\mathrm{Sym}(H)$, since by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (ii) it is the greatest lower bound of a set of nonnegative real numbers, of which $0$ is a lower bound.

Claim 1. For $x,z\in H$, condition (b) of Real Inner Product Spaces (i) read through the symmetry there, that is Elementary Identities in a Real Inner Product Space (i), gives $\langle p,x+z\rangle=\langle p,x\rangle+\langle p,z\rangle$, so

$$a(x+z)-a(x)-\langle p,z\rangle=0 .$$

Since $0\le\varepsilon\,|z|$ for every positive $\varepsilon$ and every $z$, by claim 5 of Elementary Arithmetic in an Ordered Field, any positive $\delta$ witnesses the condition of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i); hence $a$ is differentiable at every $x\in H$ with gradient $p$. The gradient map is constant, hence continuous on $H$, so $a\in C^{1}(H)$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (i). Moreover, for any positive $\rho$ and all $w,y\in H$,

$$\langle Da(x+w)-Da(x),y\rangle-0_{\mathrm{Sym}}(w,y)=\langle p-p,y\rangle-0=\langle 0_{H},y\rangle=0$$

by Elementary Identities in a Real Inner Product Space (ii), so any positive $\delta\le\rho$ witnesses the condition of The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space (i) and $0_{\mathrm{Sym}}$ is a second derivative of $a$ at $x$. The Hessian map is constant, hence continuous, so $a\in C^{2}(H)$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (ii).

Claim 2. Write $T=T_{b}$. For $x,z\in H$, the additivity and symmetry of Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i) give

$$b(x+z,x+z)=b(x,x)+b(x,z)+b(z,x)+b(z,z)=b(x,x)+2\,b(x,z)+b(z,z),$$

so that, multiplying by $2^{-1}$ and using $b(x,z)=\langle Tx,z\rangle$ from The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space (i),

$$Q(x+z)-Q(x)-\langle Tx,z\rangle=\tfrac{1}{2}\,b(z,z) .$$

By Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (ii) and claims 1 and 4 of Properties of the Absolute Value in an Ordered Field, $|\tfrac{1}{2}b(z,z)|=2^{-1}|b(z,z)|\le K\,|z|^{2}$, where $K=\lVert b\rVert\cdot 2^{-1}+1$; here $\lVert b\rVert\cdot 2^{-1}$ is nonnegative by claim 5 of Elementary Arithmetic in an Ordered Field, so $1\le K$ by claim 3 there and $0<K$ by claim 6 of Elementary Order Arithmetic in an Ordered Field together with claim 2 there.

Let $\varepsilon\in\mathbb{R}$ be positive and put $\delta=\varepsilon\,K^{-1}$, a positive real number. For $z\in H$ with $|z|<\delta$ we have $|z|\le\delta$, so two applications of claim 5 of Elementary Arithmetic in an Ordered Field, with the nonnegative multipliers $K$ and $|z|$, give

$$K\,|z|^{2}=(K\,|z|)\,|z|\le(K\,\delta)\,|z|=\varepsilon\,|z| .$$

Hence $|Q(x+z)-Q(x)-\langle Tx,z\rangle|\le\varepsilon|z|$, and $Q$ is differentiable at $x$ with gradient $Tx$. Its gradient map is $T$, which is continuous on $H$ by Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces (iii), so $Q\in C^{1}(H)$.

For the second derivative, the linearity of $T$ gives, for all $w,y\in H$,

$$\langle DQ(x+w)-DQ(x),y\rangle=\langle T(x+w)-Tx,y\rangle=\langle Tw,y\rangle=b(w,y),$$

so the difference with $b(w,y)$ is $0$ and, exactly as in claim 1, any positive $\delta\le\rho$ witnesses the condition of The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space (i). Thus $b$ is a second derivative of $Q$ at $x$; the Hessian map is constant, hence continuous, and $Q\in C^{2}(H)$.

Claim 3. By Elementary Identities in a Real Inner Product Space (v) and Real Inner Product Spaces (ii), for every $x\in H$,

$$|x-y_{0}|^{2}=\langle x,x\rangle-2\,\langle x,y_{0}\rangle+|y_{0}|^{2},$$

and $-2\langle x,y_{0}\rangle=-2\langle y_{0},x\rangle=\langle -2\,y_{0},x\rangle$ by the symmetry and condition (c) of Real Inner Product Spaces (i). Multiplying by $\tfrac{\alpha}{2}$ and using $(\alpha I)(x,x)=\alpha\,\langle x,x\rangle$ from Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv), we obtain

$$q(x)=\tfrac{1}{2}\,(\alpha I)(x,x)+\Bigl(\langle -\alpha\,y_{0},x\rangle+\tfrac{\alpha}{2}\,|y_{0}|^{2}\Bigr),$$

where $\alpha I\in\mathrm{Sym}(H)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv). The first summand is the function of claim 2 for the form $\alpha I$, and the second is the function of claim 1 with $p=-\alpha\,y_{0}$ and $c=\tfrac{\alpha}{2}|y_{0}|^{2}$; both belong to $C^{2}(H)$. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (ii), $q\in C^{2}(H)$ and, for every $x\in H$,

$$Dq(x)=T_{\alpha I}x+(-\alpha\,y_{0})=\alpha\,x-\alpha\,y_{0}=\alpha\,(x-y_{0}),\qquad D^{2}q(x)=\alpha I+0_{\mathrm{Sym}}=\alpha I ,$$

using The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space (iv) for $T_{\alpha I}x=\alpha\,x$, the distributivity axiom of the vector space $H$ together with claim 5 of Elementary Identities in a Vector Space for $\alpha x-\alpha y_{0}=\alpha(x-y_{0})$, and Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (i) for $\alpha I+0_{\mathrm{Sym}}=\alpha I$.

Claim 4. The set $W$ is open in $(H,d)$ and contained in $H$, which is itself open, and $a$, $Q$ and $q$ belong to $C^{2}(H)$ by claims 1, 2 and 3. Therefore Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space (iv) applies with $U=H$ and $U'=W$: the restrictions belong to $C^{2}(W)$, and their gradients and Hessians at every $x\in W$ are those of the unrestricted functions, namely the values given in claims 1, 2 and 3.