Example

Let X be a compact space, $\mu$ a measure on X, real or complex according as K is equal to $\mathbf{R}$ or $\mathbf{C}$, and $N : X\times X\rightarrow K$ a continuous function.

Let E denote the Banach space $\mathscr{C}(X; K)$. Suppose given a function $a\in E$ and $\lambda \in K-\{0\}$. Let us study the problem of the existence and uniqueness of the solutions $f\in E$ of the integral equation

For this, introduce the set $F_{\lambda}$ of the solutions $f\in E$ of the associated homogeneous equation

and the set $G_{\lambda}$ of the solutions $g\in E$ of the transposed homogeneous equation of (4), that is to say

If it is not empty, the set of the solutions of (3) is an affine subspace of E with direction $F_{\lambda}$.

For $f\in E$ and $x\in X$, put

the mapping $f\mapsto h(f)$ thus defined is a compact endomorphism of the Banach space E (cor. of prop. 2 of III, p. 26). The dual $E'$ of E consists of the measures on X, real or complex according as K is equal to $\mathbf{R}$ or $\mathbf{C}$ (INT, III, p. 47, § 1, n$^o3$, def. 2). The transpose $^th$ of $h$ is the endomorphism of $E'$ characterized by the relation $\langle f,^th(m)\rangle =\langle h(f), m\rangle$ for $f\in E$ and $m\in E'$, that is to say

for $m\in E'$ and $f\in E$ (INT, III, p. 84, § 4, n$^o1$, th. 2).