  The $k$ principal points of a random vector $\mathbf{X}$ are defined as a set
of points which minimize the expected squared distance between $\mathbf{X}$ and
the nearest point in the set. They are thoroughly studied in Flury (1990,
1993), Tarpey (1995) and Tarpey, Li and Flury (1995). For their treatment, the
examination is usually restricted to the family of elliptical distributions. In
this paper, we present an extension of the previous results to the functional
elliptical distribution case, i.e., when dealing with random elements over a
separable Hilbert space ${\cal H}$. Principal points for gaussian processes
were defined in Tarpey and Kinateder (2003). In this paper, we generalize the
concepts of principal points, self-consistent points and elliptical
distributions so as to fit them in this functional framework. Results linking
self-consistency and the eigenvectors of the covariance operator are
re-obtained in this new setting as well as an explicit formula for the $k=2$
case so as to include elliptically distributed random elements in ${\cal H}$.
