Let U denote a strongly continuous representation of a Lie groupG on a
Banach
space H and let dU denote the corresponding representation on the Lie algebra  g of the
group.  We define the smooth and analytic vectors for this representation. We also
define the
analytic vectors for a family of operators (unbounded) on the banach space H.  Then
we find
a
relation between the analytic vectors of the representation U and those for the
family of
operators dU(X_i) for some basis {X_i} of the Lie algebra g. Through this relation,
we try to
characterize the analytic vectors for representation of some particular groups,
specifically, the
one parameter unitary groups, the Heisenberg  groups, the group of affine
transformations and non-compact simple groups.