Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian measures on $X$ with covariance $K$ and means $x_n$ and $x$, respectively.

**Question:** How do I show that $\mathbb P_n \to \mathbb P$ weakly?

Surely this is a theorem or an exercise somewhere; e.g. in Talagrand and Ledoux's *Probability in Banach Spaces* or Vakhania, Tarieladze and Chobanyan's *Probability Distributions in Banach Spaces*.

The characteristic functions of $\mathbb P_n$ converge to those of $\mathbb P$ (simple exercise). By de Acosta's theorem, this implies that $\mathbb P_n \to \mathbb P$, provided that the family $\{\mathbb P_n\}$ is *flatly concentrated*. I'm not so familiar with the concept (hence this question), but I'm guessing this is related to the concentration of measure property of Gaussian measures.