This subject is concerned with
non-isotropic unitary spaces V over involutorial division rings D with characteristic
not 2 and with non-trivial non-archimedean exponential valuations w, which are
abelian. It will require a generalized Cauchy-Schwarz inequality relative to w. The
dimension of V over D need not be finite. Treatments of the unitary module V0
of finite vectors v in V (finite, in a technical sense), the ring L0 of linear
transformations of V that increase lengths, and the unitary group U yield
information on the normal subgroup structure of this group and the factor group
U(r)∕U(r)∩ Z, where U(r) is the r-th derived group of U and Z is the center of the
ground division ring D.
