Theorem 5

Let $\lambda \in K$.

a) The sets $F_{\lambda}$ and $G_{\lambda}$ are finite-dimensional vector subspaces of $\mathscr{C}(X; K)$ and their dimensions are equal ;

b) In order that equation (3) have at least one solution $f\in \mathscr{C}(X,K)$, it is necessary and sufficient that one have $\int_Xa(x)g(x)d\mu(x) = 0$ for every function $g\in G_{\lambda}$. The set of solutions of (3) is then an affine subspace of $\mathscr{C}(X; K)$ with direction $F_{\lambda}$;

c) One of the following conditions, which exclude each other, is satisfied :

(i) For every function $a\in \mathscr{C}(X; K)$, there exists a unique

solution $f\in \mathscr{C}(X; K)$ of the integral equation (3) ;

(ii) The homogeneous equation (4) possesses a nonzero solution