In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if
then ƒ is quadratically integrable on the real line (−∞, ∞). One may also speak of quadratic integrability over bounded intervals such as [0, 1].
The quadratically integrable functions form an inner product space whose inner product is given by
where
Since |a|2 = a a, quadratic integrability is the same as saying
It can be shown that quadratically integrable functions form a complete metric space, hence a Banach space. As we have the additional property of the inner product, this is specifically a Hilbert space. This inner product space is conventionally denoted L2.
The space of quadratically integrable functions is the Lp space in which p = 2.