Informally, an -function is a function that is square integrable, i.e.,
with respect to the measure , exists (and is finite), in which case is its L2-norm. Here is a measure space and the integral is the Lebesgue integral. The collection of functions on is called (ell-two) of L2-space, which is a Hilbert space.
On the unit interval , the functions are in for . However, the function is not in since
does not exist.
More generally, there are -complex functions obtained by replacing the absolute value of a real number in the definition with the norm of the complex number. In fact, this generalizes to functions from a measure space to any normed space.
-functions play an important role in many areas of analysis. They also arise in physics, and especially quantum mechanics, where probabilities are given as the integral of the absolute square of a wavefunction . In this and in the context of energy density, -functions arise due to the requirement that these quantities remain finite.