So let $S'(\mathbb R)$ be the space of tempered distributions, and let $\{e_n\}_{n\geq 0}$ be the family of Hermite functions (which are contained in the Schwartz space $S(\mathbb R)$). Then I define the operator $P_m:S'(\mathbb R)\to S(\mathbb R)$ by the action $$P_m\eta=\sum_{j=0}^m \langle e_j,\eta\rangle e_j$$ i.e. $P_m$ is the orthogonal projection on the span of $\{e_1,...,e_m\}$. Clearly $$\langle P_m\eta,\xi\rangle=\langle \eta,P_m\xi\rangle$$ for any $\xi,\eta\in S'(\mathbb R)$.
Is it correct to say that $P_m$ is a self-adjoint operator on $S'(\mathbb R)$? I am confused since all the definitions of self-adjointness I've found involve the use of an inner product, and my functional analysis is a bit rusty.
Thanks in advance.