I am looking for a textbook reference for a result that is something like the following:
Theorem: Let $X$, $Y$ and $\Lambda$ be TVSes with $Y,\Lambda$ in strong duality, and denote the associated bilinear map by $b$. Let $f \colon X \to \mathbb{R}$ and $g \colon X \to Y$ and define $\mathcal{L} \colon X \times \Lambda \to \mathbb{R}$ by $\mathcal{L}(x, \lambda) = f(x) + b(\lambda, g(x)-y)$. Then, $$ \inf_{x \in X}\sup_{\lambda \in \Lambda} \mathcal{L}(x,\lambda) = \sup_{\lambda \in \Lambda} \inf_{x \in X}\mathcal{L}(x,\lambda)\,.$$
Happy to make it Hilbert space based.