Square-integrable function

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

 \int_{-\infty}^\infty |f(x)|^2 \, dx < \infty,

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

 \langle f, g \rangle = \int_A f(x) \overline{g(x)} \, dx

where

Since |a|2 = a, quadratic integrability is the same as saying

 \langle f, f \rangle < \infty. \,

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.