Let $\pi$ be an admissible representation of a locally compact totally disconnected group. I have a technical problem about the proof of
$\pi$ is irreducible if and only if its contragredient is so
given in 2.15(c) of the '76 article of Bernstein and Zelevinsky. There $\pi$ is assumed to have a nontrivial proper subrepresentation $E_1$, and it is asserted that the orthogonal complement of $E_1$ be a nontrivial proper subrepresentation of the contragredient, whence the result. What I cannot figure out is the nontriviality of this orthogonal complement. We simply have to find a nonzero smooth functional which vanishes on $E_1$; this shall follow from $E_1\neq E$ (as properness of the orthogonal complement follows from $E_1\neq 0$), but how?