  We consider an enlarged dimension reduction space in functional inverse
regression. Our operator and functional analysis based approach facilitates a
compact and rigorous formulation of the functional inverse regression problem.
It also enables us to expand the possible space where the dimension reduction
functions belong. Our formulation provides a unified framework so that the
classical notions, such as covariance standardization, Mahalanobis distance,
SIR and linear discriminant analysis, can be naturally and smoothly carried out
in our enlarged space. This enlarged dimension reduction space also links to
the linear discriminant space of Gaussian measures on a separable Hilbert
space.
