In this paper we study continuous representations of locally
L
L
-analytic groups
G
G
in locally convex
K
K
-vector spaces, where
L
L
is a finite extension of
Q
p
\mathbb {Q}_p
and
K
K
is a spherically complete nonarchimedean extension field of
L
L
. The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of
G
G
, along with interesting new objects such as the action of
G
G
on global sections of equivariant vector bundles on
p
p
-adic symmetric spaces. We introduce a restricted category of such representations that we call “strongly admissible” and we show that, when
G
G
is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of
G
G
. As an application we prove the topological irreducibility of generic members of the
p
p
-adic principal series for
G
L
2
(
Q
p
)
GL_2(\mathbb {Q}_p)
. Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous
K
K
-valued representations of locally
L
L
-analytic groups.