L^2-Space

An -space on a measure space is the space of equivalence classes of measurable functions , where or , for which . Two L²-functions represent the same element when they agree almost everywhere with respect to (Axler 2026, pp. 202-203). With the L²-inner product

(1)

the -space forms a Hilbert space (Axler 2026, pp. 213 and 224). Its L²-norm is

(2)

The following bracket notation uses the same convention, linear in its first argument and antilinear in its second:

(3)

and

(4)

(5)

(6)

(7)

(8)

The inequality (8) is called Schwarz's inequality.

The basic example is when with Lebesgue measure. Another important example is when is the positive integers with counting measure, in which case it is denoted as , or "little ell-two." These are the square summable sequences (Axler 2026, pp. 194-195).

The equivalence classes make an inner product, because iff almost everywhere. Two representatives can differ on a set of measure zero without changing the L²-function.

In practice, this does not cause much trouble, except that some care has to be taken with boundary conditions in differential equations. For Lebesgue measure, the value at an individual point is not well-defined for an L²-function . For counting measure, however, each coordinate of an element is well-defined.

If an L²-function in Euclidean space with Lebesgue measure can be represented by a continuous function , then is the only continuous function representing that class. In such a case, it is not harmful to consider the L²-function as the continuous function . Also, it is often convenient to think of as the completion of the continuous functions with compact support with respect to the L²-norm.