Corollary 2

Let $G$ be a Lie group and $\pi$ an analytic linear representation of $G$ on a complete normable space $E$. Then $g \mapsto {}^t\pi(g)^{-1}$ is an analytic linear representation $\rho$ of $G$ on the complete normable space $\mathscr{L}(E, K)$\footnote{As when $K = \mathbf{R}$ or $\mathbf{C}$, the transpose ${}^t\pi(g)$ considered here is the restriction to $\mathscr{L}(E, K)$ of the transpose of $\pi(g)$ in the purely algebraic sense.} and $L(\rho)a = -{}^t(L(\pi)a)$ for all $a \in L(G)$.

This is a special case of Corollary 1.

$\rho$ is called the contragredient representation of $\pi$.

When $E$ is finite-dimensional, $L(\rho)$ is the dual representation of $L(\pi)$ in the sense of Chapter I, § 3, no. 3.