squareintegrable

A function is said to be square-integrable if the integral of the absolute value of its square over its domain is finite. Mathematically, for a real-valued function $f(x)$ defined on an interval $[a, b]$, it is square-integrable if $\int_a^b |f(x)|^2 dx < \infty$. For complex-valued functions, the absolute value is taken before squaring. The domain can also be an infinite interval or a more general measure space.

The concept of square-integrability is fundamental in various branches of mathematics and physics, particularly in functional analysis and quantum mechanics. Functions that are square-integrable form a Hilbert space, denoted as $L^2$. This space has a rich structure and is equipped with an inner product defined by $\langle f, g \rangle = \int f(x) \overline{g(x)} dx$, where $\overline{g(x)}$ is the complex conjugate of $g(x)$. The norm in this space is derived from the integral of the square of the function, making the Hilbert space property directly related to square-integrability.

In quantum mechanics, the wave function describing a particle is required to be square-integrable. This condition ensures that the probability of finding the particle somewhere in space is finite, as the integral of the square of the wave function represents the total probability. Functions that are not square-integrable cannot represent physical states because they would imply an infinite probability of finding the particle.