Closed Superlevel Sets of a Sum on a Product Space and of a Doubled Function

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E_{1}$ and $E_{2}$ be real inner product spaces, with the product $E_{1}\times E_{2}$, let $S_{1}\subseteq E_{1}$ and $S_{2}\subseteq E_{2}$ be nonempty, and let $u_{1}:S_{1}\to\mathbb{R}$ and $u_{2}:S_{2}\to\mathbb{R}$ be bounded above and have closed superlevel sets in $E_{1}$ and in $E_{2}$ respectively. Then the following hold.

1. (Sums on a product)(i) The function $S_{1}\times S_{2}\to\mathbb{R}$ whose value at $(x,y)$ is $u_{1}(x)+u_{2}(y)$ has closed superlevel sets in $E_{1}\times E_{2}$.

2. (The doubled function)(ii) Assume in addition that $E_{1}=E_{2}=H$ for a real Hilbert space $H$, let $\alpha\in\mathbb{R}$, and let $\Phi:S_{1}\times S_{2}\to\mathbb{R}$ be the function with value

$$\Phi(x,y)=u_{1}(x)+u_{2}(y)-\tfrac{\alpha}{2}\,|x-y|_{H}^{2}$$

at $(x,y)$. Then $\Phi$ has closed superlevel sets in $H\times H$. If moreover $0\le\alpha$ and there are $C_{1},C_{2}\in\mathbb{R}$ and a positive $\kappa\in\mathbb{R}$ with

$$u_{1}(x)\le C_{1}-\kappa|x|_{H}^{2}\ \text{ for every }x\in S_{1},\qquad u_{2}(y)\le C_{2}-\kappa|y|_{H}^{2}\ \text{ for every }y\in S_{2},$$

then

$$\Phi(x,y)\ \le\ C_{1}+C_{2}-\kappa\,|(x,y)|^{2}\qquad\text{for every }(x,y)\in S_{1}\times S_{2},$$

where $|(x,y)|$ is the norm of $H\times H$ and the identity $|(x,y)|^{2}=|x|_{H}^{2}+|y|_{H}^{2}$ is recorded in Properties of the Product of Two Real Inner Product Spaces (ii).