The Doubling Form on the Product of a Real Hilbert Space with Itself

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and let $H\times H$ be its product with itself, which is a real inner product space with norm $|\cdot|$ and distance $d$ by Properties of the Product of Two Real Inner Product Spaces (i) and a real Hilbert space by Properties of the Product of Two Real Inner Product Spaces (vi). Let $\alpha\in\mathbb{R}$. For $b$ in the set $\mathrm{Sym}(H\times H)$ of bounded symmetric bilinear forms on $H\times H$, with norm $\lVert\cdot\rVert$, the operator represented by $b$ is written $T_{b}$. The classes $C^{2}(W)$, the gradient and the Hessian are as defined there. Then the following hold.

1. (The doubling form)(i) The map $b$ assigning to each pair of elements $(x_{1},x_{2})$ and $(y_{1},y_{2})$ of $H\times H$ the real number

$$b\bigl((x_{1},x_{2}),(y_{1},y_{2})\bigr)=\langle x_{1}-x_{2},\,y_{1}-y_{2}\rangle$$

belongs to $\mathrm{Sym}(H\times H)$ and satisfies $\lVert b\rVert\le 2$, and

$$T_{b}(x_{1},x_{2})=(x_{1}-x_{2},\,x_{2}-x_{1})\qquad\text{for every }(x_{1},x_{2})\in H\times H .$$

Moreover $b\bigl((x_{1},x_{2}),(x_{1},x_{2})\bigr)=|x_{1}-x_{2}|^{2}$.

2. (The squared distance between the coordinates)(ii) Let $q:H\times H\to\mathbb{R}$ be given by

$$q(x_{1},x_{2})=\tfrac{\alpha}{2}\,|x_{1}-x_{2}|^{2}.$$

Then $q\in C^{2}(H\times H)$ and, for every $(x_{1},x_{2})\in H\times H$,

$$Dq(x_{1},x_{2})=\alpha\,(x_{1}-x_{2},\,x_{2}-x_{1}), \qquad D^{2}q(x_{1},x_{2})=\alpha\,b .$$

3. (Restriction to an open subset)(iii) Let $W\subseteq H\times H$ be open in $(H\times H,d)$. Then the restriction of $q$ to $W$ belongs to $C^{2}(W)$, with the gradient and Hessian given in claim 2 at every point of $W$.