I am reading about cylindrical brownian motion in the monograph of Prato and Zabczyk. For this construction a Hilbert-Schmidt operator is used, between to separable Hilbert spaces $U$ and $U_1.$ Let $J$ denote this operator, and let $\{g_j\}$ be an orthonormal basis in $U_0,$ which is another separable Hilbert space which is embedded in $U$. The whole point of the construction, as it seems to me, is that this latter embedding may well be the identity.
After having constructed a cylindrical Brownian motion, the following inequality is taken (almoast) for granted, for any $a \in U$:
$$\sum_{j \ge 1} |<a, g_j>|^2 \le \sum_{j \ge 1} ||a||^2 ||g_j||^2 \le C ||J||^2 ||a||^2 \sum_{j \ge 1} ||Jg_i||_1^2 < +\infty$$
Here the norm without pedex is the norm in $U.$
Not only I don't know how to prove the central inequality, but to me it seems completely wrong (for example if $U_0 = U)$.
Could anyone clear my thoughts?