The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, let $\mathcal{P}_{2}(\mathbb{R}^{d})$ be the set of probability measures on $\mathbb{R}^{d}$ with finite second moment, and let $W_{2}$ be the quadratic Wasserstein distance, a nonnegative real-valued function on pairs of elements of $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then claims 1 to 3 hold for all $\mu,\nu,\lambda\in\mathcal{P}_{2}(\mathbb{R}^{d})$, 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 $\mathcal{P}_{2}(\mathbb{R}^{d})$; the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ is called the quadratic Wasserstein space over $\mathbb{R}^{d}$.