In mathematics, a square-integrable function, also called a quadratically integrable function or ${\displaystyle L^{2}}$ function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line ${\displaystyle (-\infty ,+\infty )}$ is defined as follows.

${\displaystyle f:\mathbb {R} \to \mathbb {C} {\text{ square integrable}}\quad \iff \quad \int _{-\infty }^{\infty }|f(x)|^{2}\,\mathrm {d} x<\infty }$

One may also speak of quadratic integrability over bounded intervals such as ${\displaystyle [a,b]}$ for ${\displaystyle a\leq b}$.

${\displaystyle f:[a,b]\to \mathbb {C} {\text{ square integrable on }}[a,b]\quad \iff \quad \int _{a}^{b}|f(x)|^{2}\,\mathrm {d} x<\infty }$

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part.

The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the ${\displaystyle L^{p}}$ space with ${\displaystyle p=2.}$ Among the ${\displaystyle L^{p}}$ spaces, the class of square integrable functions is unique in being compatible with an inner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a Hilbert space, since all of the ${\displaystyle L^{p}}$ spaces are complete under their respective ${\displaystyle p}$-norms.

Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal almost everywhere.

The square integrable functions (in the sense mentioned in which a "function" actually means an equivalence class of functions that are equal almost everywhere) form an inner product space with inner product given by $${\displaystyle \langle f,g\rangle =\int _{A}f(x){\overline {g(x)}}\,\mathrm {d} x,}$$ where