# Bilinear integral form

$$J ( \phi , \psi ) = \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K(x, s) \phi (x) \overline{ {\psi (s) }}\; dx ds ,$$
where $K(x, s)$ is a given (usually complex-valued) square-integrable function of real variables, and $\phi (x)$, $\psi (x)$ are arbitrary (also complex-valued) square-integrable functions, while $\overline{ {\psi (s) }}\;$ is the complex conjugate function of $\psi (s)$. If $\psi (s) = \phi (s)$, $J( \phi , \phi )$ is said to be a quadratic integral form.