The Quadratic Wasserstein Distance is a Metric on the 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 $\mathcal{P}_{2}(X)$ be the set of probability measures on $X$ with finite second moment and $W_{2}$ the quadratic Wasserstein distance, a nonnegative real-valued function on pairs of elements of $\mathcal{P}_{2}(X)$. Then claims 1 to 3 hold for all $\mu,\nu,\lambda\in\mathcal{P}_{2}(X)$, and claim 4 holds.

1. (Symmetry)¶ $W_{2}(\mu,\nu)=W_{2}(\nu,\mu)$.

2. (Separation)¶ $W_{2}(\mu,\nu)=0$ if and only if $\mu=\nu$.

3. (Triangle inequality)¶ $W_{2}(\mu,\lambda)\le W_{2}(\mu,\nu)+W_{2}(\nu,\lambda)$.

4. (The Wasserstein space)¶ $W_{2}$ is a metric on $\mathcal{P}_{2}(X)$; the metric space $(\mathcal{P}_{2}(X),W_{2})$ is called the quadratic Wasserstein space over $X$.