TOPICS
Search

L^2-Function


Informally, an L^2-function is a function f:X->R that is square integrable, i.e.,

 |f|^2=int_X|f|^2dmu

with respect to the measure mu, exists (and is finite), in which case |f| is its L2-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 L2-space, which is a Hilbert space.

L2-Function

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

 int_0^1(x^(-1/2))^2dx=int_0^1(dx)/x

does not exist.

More generally, there are L^2-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 X to any normed space.

L^2-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 psi. In this and in the context of energy density, L^2-functions arise due to the requirement that these quantities remain finite.


See also

Hilbert Space, Lebesgue Integral, L-p-Space, L2-Space, Measure, Measure Space, Square Integrable

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "L^2-Function." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/L2-Function.html

Subject classifications