Let $F$ be a Banach space and $L(F)$ be the set of all its bounded linear
operators. In this note, we are interested in the asymptotic behavior
(recurrence and chain recurrence) of the solution of the following initial
value problem \label{eqlinear} x'(t) = X(t)x(t), \qquad x(0) = x, where $x \in
F$ and the map $t \mapsto X(t) \in L(F)$ is a $T$-periodic continuous curve.
This asymptotic behavior is related to the asymptotic behavior of the
discrete-time flow on $F$ generated by the invertible operator $g \in L(F)$
given by the associated fundamental solution at time $T$.