Proposition 3

Let $(V, \pi )$ be a finite-dimensional linear representation of the algebra A. The kernel $\mathfrak{a}$ of $\pi$ is a two-sided ideal of A of finite codimension, and $\Theta_{\pi}(A)$ is the orthogonal of $\mathfrak{a}$ in $A^*$. The transpose of the mapping $\pi$ defines an isomorphism from the dual of the K-vector space $\pi (A)$ to $\Theta_{\pi}(A)$.

Since End$_K(V)$ is finite-dimensional over $K,\mathfrak{a}$ is a two-sided ideal of A of finite codimension. The vector space $\Theta_{\pi}(A)$ is a finite-dimensional subspace of $A^*$, and its orthogonal in A is equal to $\mathfrak{a}$; by Theorem 7 of II, §7, No. 5, p. 301, the space $\Theta_{\pi}(A)$ is therefore the orthogonal of $\mathfrak{a}$ in $A^*$. Moreover, since $\mathfrak{a}$ is the kernel of $\pi$, the transpose of the mapping $\pi$ defines an isomorphism from the dual of $\pi (A)$ to the orthogonal of $\mathfrak{a}$ in $A^*$ (II, §7, No. 5, p. 299, Corollary of Theorem 5).