  In this paper, we prove that any relative character (a.k.a. spherical
character) of any admissible representation of a real reductive group with
respect to any pair of spherical subgroups is a holonomic distribution on the
group. This implies that the restriction of the relative character to an open
dense subset is given by an analytic function. The proof is based on an
argument from algebraic geometry and thus implies also analogous results in the
p-adic case.
  As an application, we give a short proof of some results from [KO13,KS16] on
boundedness and finiteness of multiplicities of irreducible representations in
the space of functions on a spherical space.
  In order to deduce this application we prove relative and quantitative
analogs of the Bernstein-Kashiwara theorem, which states that the space of
solutions of a holonomic system of differential equations in the space of
distributions is finite-dimensional. We also deduce that, for every algebraic
group $G$ defined over $\mathbb{R}$, the space of $G(\mathbb{R})$-equivariant
distributions on the manifold of real points of any algebraic $G$-manifold $X$
is finite-dimensional if $G$ has finitely many orbits on $X$.
