  In this paper we develop two types of tools to deal with differentiability
properties of vectors in continuous representations $\pi \: G \to \GL(V)$ of an
infinite dimensional Lie group $G$ on a locally convex space $V$. The first
class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a
Banach--Lie group, we define a topology on the space $V^\infty$ of smooth
vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach
space, then $V^\infty$ is a Fr\'echet space. This applies in particular to
$C^*$-dynamical systems $(\cA,G, \alpha)$, where $G$ is a Banach--Lie group.
For unitary representations we show that a vector $v$ is smooth if the
corresponding positive definite function $\la \pi(g)v,v\ra$ is smooth.
  The second class of results concerns criteria for $C^k$-vectors in terms of
operators of the derived representation for a Banach--Lie group $G$ acting on a
Banach space $V$. In particular, we provide for each $k \in \N$ examples of
continuous unitary representations for which the space of $C^{k+1}$-vectors is
trivial and the space of $C^k$-vectors is dense.
