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Let $G$ be a real linear reductive group and $K$ be a maximal compactsubgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Liealgebra $\mathfrak{p}$, and $\mathfrak{n}$ be its nilradical. In this paper weshow that: for any admissible finitely generated moderate growth smoothFr\'echet representation $V$ of $G$, the inclusion $V_{K}\subset V$ inducesisomorphisms $H_{i}(\mathfrak{n},V_{K})\cong H_{i}(\mathfrak{n},V)$ ($i\geq0$), where $V_{K}$ denotes the $(\mathfrak{g},K)$ module of $K$ finite vectorsin $V$. This is called Casselman's comparison theorem. As a consequence, weshow that: for any $k\geq 1$, $\mathfrak{n}^{k}V$ is a closed subspace of $V$and the inclusion $V_{K}\subset V$ induces an isomorphism$V_{K}/\mathfrak{n}^{k}V_{K}= V/\mathfrak{n}^{k}V$. This strengthensCasselman's automatic continuity theorem.

A proof of Casselman's comparison theorem for standard minimal parabolic subalgebra