I'm reading Bump's book Automorphic Forms and Representations and getting stuck in Section 4.2 about contragredient representation of smooth representations.
Setup: Let $G$ be a locally profinite group (i.e. totally disconneted, Hausdorff and locally compact, e.g. $\mathrm{GL}_n(\mathbb{Q}_p)$) and $K$ be its compact open subgroup. Let $(\pi, V)$ be a smooth representation of $G$. Then by Bump's Proposition 4.2.2, we see $$ V = \oplus_{\rho \in \widehat{K}} V(\rho), $$ where $\widehat{K}$ is the set of equivalent classes of irreducible smooth representations of $K$ (which are automatically of finite dimension with open kernel) and $V(\rho)$ is the $\rho$-isotypic part of $V$. Then $$ V^{\ast} = \prod_{\rho \in \widehat{K}} (V(\rho))^{\ast}, \quad (\star) $$ where $(-)^{\ast}$ is the linear dual of $-$.
Problem for me: Then to isolate the smooth linear functionals in $V^{\ast}$, Bump claimed that it is not difficult to see that a linear functional $\lambda \in V^{\ast}$ is smooth if and only if $\lambda$ is zero on all but finitely many of the summands in $(\star)$.
My question: How to show this claim?
My attempts: I have managed to show the $(\Leftarrow)$ direction. Indeed, note that a linear functional $\lambda$ is smooth if and only if there exist an open subgroup $H$ of $G$ such that $\lambda = \lambda \circ \pi(h)$ for any $h \in H$. Now for $\lambda$ vanishing on all but finitely many summands $\rho_1, \ldots, \rho_k$, taking $H := \cap_{i=1}^{k} \ker \rho_i$ does the trick.
BUT, how to show the $(\Rightarrow)$ direction?
Thank you all for answering or commenting! :)