Corollary

The restricted dual Θ(A) of A is the set of coefficients of the finite-dimensional linear representations of A.

By definition, Θ(A) is the union of the orthogonals of the two-sided ideals of A of finite codimension. We therefore have $\Theta_{\pi}(A)\subset \Theta (A)$ for every finite-dimensional linear representation $\pi$ of A. Conversely, let $f$ be an element of Θ(A), and let $\mathfrak{a}$ be a two-sided ideal of finite codimension, contained in the kernel of $f$ (VIII, p. 376, Definition 2). Denote by $\pi$ the linear representation of A in $A/\mathfrak{a}$ deduced from the left regular representation in A by passing to the quotient. Let $x$ be the class of 1 (mod $\mathfrak{a}$), and let $x^*$ be the linear form on $A/\mathfrak{a}$ deduced from $f$. We have $f=c_{\pi}(x, x^*)$, so that $f$ is a coefficient of $\pi$.

Take note that if $(V, \pi )$ is a linear representation of A that is not finite-dimensional over K, then the space $\Theta_{\pi}(A)$ is not necessarily contained in Θ(A).