  The purpose of this paper is to prove an equivariant Riemann-Roch theorem for
schemes or algebraic spaces with an action of a linear algebraic group $G$. For
a $G$-space $X$, this theorem gives an isomorphism between a completion of the
equivariant Grothendieck group and a completion of equivariant equivariant Chow
groups.
  The key to proving this isomorphism is a geometric description of completions
of the equivariant Grothendieck group. Besides Riemann-Roch, this result has
some purely $K$-theoretic applications. In particular, we prove a conjecture of
K\"ock (in the case of regular schemes) and extend to arbitrary characteristic
a result of Segal on representation rings.
