  The lack of a $p$-adic Haar measure causes many methods of traditional
representation theory to break down when applied to continuous representations
of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic
field $K$. For example, Diarra showed that the abelian group $G=\dZ$ has an
enormous wealth of infinite dimensional, topologically irreducible Banach space
representations.
  We therefore address the problem of finding an additional ''finiteness''
condition on such representations that will lead to a reasonable theory. We
introduce such a condition that we call ''admissibility''. We show that the
category of all admissible $G$-representations is reasonable -- in fact, it is
abelian and of a purely algebraic nature -- by showing that it is
anti-equivalent to the category of all finitely generated modules over a
certain kind of completed group ring $K[[G]]$.
  As an application of our methods we determine the topological irreducibility
as well as the intertwining maps for representations of $GL_2(\dZ)$ obtained by
induction of a continuous character from the subgroup of lower triangular
matrices.
