Let
G
G
be a real linear reductive group and
K
K
be a maximal compact subgroup. Let
P
P
be a minimal parabolic subgroup of
G
G
with complexified Lie algebra
p
\mathfrak {p}
, and
n
\mathfrak {n}
be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation
V
V
of
G
G
, the inclusion
V
K
⊂
V
V_{K}\subset V
induces isomorphisms
H
i
(
n
,
V
K
)
≅
H
i
(
n
,
V
)
H_{i}(\mathfrak {n},V_{K})\cong H_{i}(\mathfrak {n},V)
(
i
≥
0
i\geq 0
), where
V
K
V_{K}
denotes the
(
g
,
K
)
(\mathfrak {g},K)
module of
K
K
finite vectors in
V
V
. This is called Casselman’s comparison theorem (see Henryk Hecht and Joseph L. Taylor [A remark on Casselman’s comparison theorem, Birkhäuser Boston, Boston, Ma, 1998, pp. 139–146]). As a consequence, we show that: for any
k
≥
1
k\geq 1
,
n
k
V
\mathfrak {n}^{k}V
is a closed subspace of
V
V
and the inclusion
V
K
⊂
V
V_{K}\subset V
induces an isomorphism
V
K
/
n
k
V
K
=
V
/
n
k
V
V_{K}/\mathfrak {n}^{k}V_{K}= V/\mathfrak {n}^{k}V
. This strengthens Casselman’s automatic continuity theorem (see W. Casselman [Canad. J. Math. 41 (1989), pp. 385–438] and Nolan R. Wallach [Real reductive groups, Academic Press, Boston, MA, 1992]).