  Let ${\mbox{$\mbox{\boldmath $f$}$}}$ be a square-integrable, zero-mean,
random vector with observable realizations in a Hilbert space $H$, and let
${\mbox{$\mbox{\boldmath $g$}$}}$ be an associated square-integrable,
zero-mean, random vector with realizations, which are not observable, in a
Hilbert space $K$. We seek an optimal filter in the form of a closed linear
operator $X$ acting on the observable realizations of a proximate vector
${\mbox{$\mbox{\boldmath $f$}$}}_{\epsilon} \approx {\mbox{$\mbox{\boldmath
$f$}$}}$ that provides the best estimate $\widehat{{\mbox{$\mbox{\boldmath
$g$}$}}}_{\epsilon} = X {\mbox{$\mbox{\boldmath $f$}$}}_{\epsilon}$ of the
vector ${\mbox{$\mbox{\boldmath $f$}$}}$. We assume the required covariance
operators are known. The results are illustrated with a typical example.
