  We combine the geometric realization of principal series representations of
the author, with the Bott--Borel--Weil Theorem for direct limits of compact
groups of Natarajan, Rodriguez-Carrington and the author, obtaining limits of
principal series representations for direct limits of real reductive Lie
groups. We introduce the notion of weakly parabolic direct limits and relate it
to the conditions that the limit representations are norm--preserving
representations on a Banach space or unitary representations on a Hilbert
space. We specialize the results to diagonal embedding direct limit groups.
Finally we discuss the possibilities of extending the results to limits of
tempered series other than the principal series.
