Informally, an L^2-function is a function f:X->R that is square integrable, i.e., ( left bracketing bar f right bracketing bar )^2 = integral_X ( left bracketing bar f right bracketing bar )^2 d μ with respect to the measure μ, exists (and is finite), in which case left bracketing bar f right bracketing bar is its L^2-norm. Here X is a measure space and the integral is the Lebesgue integral. The collection of L^2 functions on X is called L^2(X) (ell-two) of L^2-space, which is a Hilbert space. On the unit interval (0, 1), the functions f(x) = 1/x^p are in L^2 for p<1/2. However, the function f(x) = x^(-1/2) is not in L^2 since integral_0^1 (x^(-1/2))^2 d x = integral_0^1 (d x)/x does not exist.

Hilbert space | L^2-space | Lebesgue integral | L^p-space | measure | measure space | square integrable