Existence of an Optimal Coupling of Two Borel Probability Measures with Finite Second Moment on a Hilbert Space

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, let $\mu,\nu$ belong to the set $\mathcal{P}_{2}(X)$ of probability measures with finite second moment, and let $W_{2}$ be the quadratic Wasserstein distance.

¶ There is a coupling $\pi\in\Pi(\mu,\nu)$ whose quadratic cost satisfies $I(\pi)=W_{2}(\mu,\nu)^{2}$.