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Inmathematics, asquare-integrable function, also called aquadratically integrable function or${\displaystyle L^2}$ function orsquare-summable function,^[1] is areal- orcomplex-valuedmeasurable function for which theintegral of the square of theabsolute value is finite. Thus, square-integrability on the real line${\displaystyle (-\mathrm{\infty },+\mathrm{\infty })}$ is defined as follows.

One may also speak of quadratic integrability over bounded intervals such as${\displaystyle [a,b]}$ for${\displaystyle a\le b}$.^[2]

An equivalent definition is to say that the square of the function itself (rather than of its absolute value) isLebesgue 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.

Thevector space of (equivalence classes of) square integrable functions (with respect toLebesgue 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 aninner product, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form aHilbert space, since all of the${\displaystyle L^p}$ spaces arecomplete 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 equalalmost everywhere.

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

• ${\displaystyle f}$ and${\displaystyle g}$ are square integrable functions,
• ${\displaystyle \overline{f(x)}}$ is thecomplex conjugate of${\displaystyle f(x),}$
• ${\displaystyle A}$ is the set over which one integrates—in the first definition (given in the introduction above),${\displaystyle A}$ is${\displaystyle (-\mathrm{\infty },+\mathrm{\infty })}$, in the second,${\displaystyle A}$ is${\displaystyle [a,b]}$.

Since${\displaystyle |a{|}^2=a\cdot \overline{a}}$, square integrability is the same as saying

It can be shown that square integrable functions form acomplete metric space under the metric induced by the inner product defined above.A complete metric space is also called aCauchy space, because sequences in such metric spaces converge if and only if they areCauchy.A space that is complete under the metric induced by a norm is aBanach space.Therefore, the space of square integrable functions is a Banach space, under the metric induced by the norm, which in turn is induced by the inner product.As we have the additional property of the inner product, this is specifically aHilbert space, because the space is complete under the metric induced by the inner product.

This inner product space is conventionally denoted by${\displaystyle \left(L_2,⟨\cdot ,\cdot {⟩}_2\right)}$ and many times abbreviated as${\displaystyle L_2.}$Note that${\displaystyle L_2}$ denotes the set of square integrable functions, but no selection of metric, norm or inner product are specified by this notation.The set, together with the specific inner product${\displaystyle ⟨\cdot ,\cdot {⟩}_2}$ specify the inner product space.

The space of square integrable functions is the${\displaystyle L^p}$ space in which${\displaystyle p=2.}$

The function${\displaystyle \frac{1}{x^n},}$ defined on${\displaystyle (0,1),}$ is in${\displaystyle L^2}$ for but not for${\displaystyle n=\frac{1}{2}.}$^[1] The function${\displaystyle \frac{1}{x},}$ defined on${\displaystyle [1,\mathrm{\infty }),}$ is square-integrable.^[3]

Bounded functions, defined on${\displaystyle [0,1],}$ are square-integrable. These functions are also in${\displaystyle L^p,}$ for any value of${\displaystyle p.}$^[3]

The function${\displaystyle \frac{1}{x},}$ defined on${\displaystyle [0,1],}$ where the value at${\displaystyle 0}$ is arbitrary. Furthermore, this function is not in${\displaystyle L^p}$ for any value of${\displaystyle p}$ in${\displaystyle [1,\mathrm{\infty }).}$^[3]

• Inner product space
• ${\displaystyle L^p}$ space – Function spaces generalizing finite-dimensional p norm spaces

• Functional analysis
• Mathematical analysis
• Lp spaces

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