  The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of
the equivalent classes of curves on $M$ that agree up to their acceleration. It
is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$
becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear
connection. Here we extend this result to $M$ modeled on an arbitrarily chosen
Banach space and more generally to those Fr\'{e}chet manifolds which can be
obtained as projective limits of Banach manifolds. The result may have
application in the study of infinite-dimensional dynamical systems.
