  We give a generalization of Kostant's theorem on Lie algebra cohomology of
finite dimensional highest weight representations to some infinite dimensional
cases over a $p$-adic family of highest weight distribution representations.
For proving this, we develop a theory of eigen orthonormalizable Banach
representations of $p$-adic torus over an affinoid algebra, and we construct an
eigen orthonormalizable weight completion of the distribution representations.
