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

Each result cited is universally quantified over the data in its own statement. Let $K_{1},K_{2}\in\mathbb{R}$ satisfy $u_{1}(x)\le K_{1}$ for every $x\in S_{1}$ and $u_{2}(y)\le K_{2}$ for every $y\in S_{2}$, and write $u(x,y)=u_{1}(x)+u_{2}(y)$ for $(x,y)\in S_{1}\times S_{2}$.

Claim 1. We verify the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i) in the metric space $E_{1}\times E_{2}$. Let $\bigl((x_{m},y_{m})\bigr)_{m\in\mathbb{N}}$ be a sequence in $S_{1}\times S_{2}$ converging in $E_{1}\times E_{2}$ to a point $(x,y)$, and let $t\in\mathbb{R}$ satisfy $t\le u(x_{m},y_{m})$ for every $m\in\mathbb{N}$. By Properties of the Product of Two Real Inner Product Spaces (iv) the sequence $(x_{m})_{m\in\mathbb{N}}$ converges to $x$ in $E_{1}$ and $(y_{m})_{m\in\mathbb{N}}$ converges to $y$ in $E_{2}$.

Since $u_{2}(y_{m})\le K_{2}$, we get $t-K_{2}\le u_{1}(x_{m})$ for every $m$, so Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i), applied to $u_{1}$, gives $x\in S_{1}$; symmetrically $y\in S_{2}$, so $(x,y)\in S_{1}\times S_{2}$.

By Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (ii) the function $u_{1}$ is upper semicontinuous on $S_{1}$ and $u_{2}$ is upper semicontinuous on $S_{2}$. Let $\epsilon\in\mathbb{R}$ be positive. By upper semicontinuity at $x$ relative to $S_{1}$ there is a positive $\rho_{1}\in\mathbb{R}$ such that every $x'\in S_{1}$ with $d_{E_{1}}(x',x)<\rho_{1}$ satisfies $u_{1}(x')\le u_{1}(x)+\epsilon$, and similarly there is a positive $\rho_{2}$ for $u_{2}$ at $y$. By the convergence of the two coordinate sequences there is $m\in\mathbb{N}$ with $d_{E_{1}}(x_{m},x)<\rho_{1}$ and $d_{E_{2}}(y_{m},y)<\rho_{2}$, and for that $m$

$$t\le u_{1}(x_{m})+u_{2}(y_{m})\le u_{1}(x)+u_{2}(y)+2\epsilon .$$

As $\epsilon$ ranges over all positive reals so does $2\epsilon$, so Comparison of Real Numbers with Arbitrary Positive Slack gives $t\le u_{1}(x)+u_{2}(y)=u(x,y)$. By Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (i) again, $u$ has closed superlevel sets in $E_{1}\times E_{2}$.

Claim 2. Let $q:H\times H\to\mathbb{R}$ be given by $q(x,y)=\tfrac{\alpha}{2}|x-y|_{H}^{2}$. By The Doubling Form on the Product of a Real Hilbert Space with Itself (ii) the function $q$ belongs to $C^{2}(H\times H)$; in particular it is differentiable at every point of $H\times H$, hence continuous on $H\times H$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space (ii). Since $\Phi(x,y)=u(x,y)-q(x,y)$ for $(x,y)\in S_{1}\times S_{2}$, claim 1 together with Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (iii) shows that $\Phi$ has closed superlevel sets in $H\times H$.

Assume now the stated bounds and $0\le\alpha$. For $(x,y)\in S_{1}\times S_{2}$ we have $0\le\tfrac{\alpha}{2}|x-y|_{H}^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, hence

$$\Phi(x,y)\le u_{1}(x)+u_{2}(y)\le C_{1}+C_{2}-\kappa\bigl(|x|_{H}^{2}+|y|_{H}^{2}\bigr)=C_{1}+C_{2}-\kappa|(x,y)|^{2},$$

the last equality by the formula for the norm of the product in Properties of the Product of Two Real Inner Product Spaces (ii).