Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with distribution $\mathbb P$.

I would like a simple upper bound on the size of $\mathbb E \|x\|^2$ in terms of the operator norm $\|K\|$. In his textbook *The Concentration of Measure Phenomenon*, Ledoux proves that $$\mathbb E\|x\|^2 \le 4 \|K\|$$ as a consequence of a concentration inequality for a simpler example (where the vectors are sums of vectors with i.i.d. random coefficients $\eta_i$ such that $|\eta_i| \le 1$ a.s.). Because of the boundedness assumption, the argument doesn't quite work in this setting, though I'm certain I could generalize it if I needed to.

Nonetheless, I'm certain that the estimate I'm looking for is buried somewhere in the literature on concentration of measure (and in fact is probably due to Talagrand). Could you please point me in the right direction?

**Edit:** The inequality as I previously wrote it is incorrect. The correct inequality should be $$\operatorname{Var}(\|x\|) \le 4\|K\|,$$
implying $$\mathbb E\|x\|^2 \le 4\|K\| + \left( \mathbb E\|x\| \right)^2.$$ That is, the size of typical random element could be quite large (i.e. $\mathbb E\|x\| \gg 1$, as in Mark Meckes's example in the comments), but the deviation is only of the order $\|K\|^{1/2}$.