Completeness of the Quadratic Wasserstein Space and of the Noise Wasserstein Space over a Hilbert Space

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $(\mathcal{P}_{2}(X),W_{2})$ be the quadratic Wasserstein space over $X$ and $(\mathcal{P}^{a}_{\rho},W_{a})$ the noise Wasserstein space based at $\rho$, with $\mathcal{P}^{a}_{\rho}$ the set of The Measures Noise-Connected to the Reference Measure §space, which are metric spaces by The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §metric and The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric. Then the following hold.

1. (The quadratic Wasserstein space)¶ $(\mathcal{P}_{2}(X),W_{2})$ is a complete metric space: every Cauchy sequence in it converges to an element of $\mathcal{P}_{2}(X)$.

2. (The noise Wasserstein space)¶ $(\mathcal{P}^{a}_{\rho},W_{a})$ is a complete metric space: every Cauchy sequence in it converges to an element of $\mathcal{P}^{a}_{\rho}$.