Penalty Pairs with Closed Score

In the setting of Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in the space $L^{2}(\nu;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{\nu}$; for $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\nu$, the composition $\Sigma(\nu)\circ X$ is the element of $L^{2}(\Omega;\mathbb{R}^{d})$ of that clause, which satisfies $\lVert\Sigma(\nu)\circ X\rVert_{L^{2}}=\lVert\Sigma(\nu)\rVert_{\nu}$. Convergence of a sequence in $L^{2}(\Omega;\mathbb{R}^{d})$ is that of the metric $d_{L^{2}}$ fixed in Plans, Marginals, Vector Fields and Symmetric Matrices on the Wasserstein Space: Standing Notation §dimensions, and weak convergence of a sequence in the real inner product space $L^{2}(\Omega;\mathbb{R}^{d})$ is that of that definition.

(Closed score)¶ The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ has closed score if the following holds for every nonnegative $R\in\mathbb{R}$. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence in $L^{2}(\Omega;\mathbb{R}^{d})$ such that $\mathcal{L}(X_{n})\in\mathcal{D}_{\Sigma}$ and

for every $n\in\mathbb{N}$, and suppose that $(X_{n})_{n\in\mathbb{N}}$ converges to $X$ in $L^{2}(\Omega;\mathbb{R}^{d})$. Then $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$, and the sequence whose $n$-th term is $\Sigma(\mathcal{L}(X_{n}))\circ X_{n}$ converges weakly to $\Sigma(\mathcal{L}(X))\circ X$.

The condition asks of the score exactly what closedness asks of an unbounded operator: along a sequence on which it is bounded and whose arguments converge, the limit point lies in its domain and the values converge weakly to the value at the limit. No continuity of $\nu\mapsto\Sigma(\nu)$ is asked, and none holds for the scores of interest; the uniform bound $R$ is the one a coercivity hypothesis on an equation operator is expected to supply at the data of a doubling.