  This paper provides a mathematical framework for Fisher information analysis
for inverse problems based on Gaussian noise on infinite-dimensional Hilbert
space. The covariance operator for the Gaussian noise is assumed to be trace
class, and the Jacobian of the forward operator Hilbert-Schmidt. We show that
the appropriate space for defining the Fisher information is given by the
Cameron-Martin space. This is mainly because the range space of the covariance
operator always is strictly smaller than the Hilbert space. For the Fisher
information to be well-defined, it is furthermore required that the range space
of the Jacobian is contained in the Cameron-Martin space. In order for this
condition to hold and for the Fisher information to be trace class, a
sufficient condition is formulated based on the singular values of the Jacobian
as well as of the eigenvalues of the covariance operator, together with some
regularity assumptions regarding their relative rate of convergence. An
explicit example is given regarding an electromagnetic inverse source problem
with "external" spherically isotropic noise, as well as "internal" additive
uncorrelated noise.
