Let $(Z,\|\cdot\|_{Z})$ be a separable Banach space and let $(H,\|\cdot\|_{H})$ be a separable Hilbert space. Let $T:Z\to H$ be a continuous linear operator. Is it true that $T(A)$ is Borel measurable for all open sets $A$?
Equivalently: is $T(B)$, the image of the unit ball $B=\{\|z\|\le1\}\subset Z$, Borel measurable in $H$?
I'm only interested in the infinite-dimensional case, $\dim(Z)=+\infty$, $\dim(H)=+\infty$, as the answer is trivial otherwise.