Square-integrable function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a square-integrable function, also called a quadratically integrable function or 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L^{2}}
 
 function or square-summable function, is a <a href="/facts/Real_number/R02gw5Pb">real</a>- or <a href="/facts/Complex_number/2w7mqI7e">complex</a>-valued <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> for which the <a href="/facts/Integral/V7lyV997">integral</a> of the square of the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> is finite. Thus, square-integrability on the real line 
 
 
 
 (
 −
 ∞
 ,
 +
 ∞
 )
 
 
 {\displaystyle (-\infty ,+\infty )}
 
 is defined as follows.

f
 :
 
 R
 
 →
 
 C
 
 
  square integrable
 
 
 
 ⟺
 
 
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 
 |
 
 f
 (
 x
 )
 
 
 |
 
 
 2
 
 
 
 
 d
 
 x
 <
 ∞
 
 
 {\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 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 for 
 
 
 
 a
 ≤
 b
 
 
 {\displaystyle a\leq b}
 
.

f
 :
 [
 a
 ,
 b
 ]
 →
 
 C
 
 
  square integrable on 
 
 [
 a
 ,
 b
 ]
 
 
 ⟺
 
 
 
 ∫
 
 a
 
 
 b
 
 
 
 |
 
 f
 (
 x
 )
 
 
 |
 
 
 2
 
 
 
 
 d
 
 x
 <
 ∞
 
 
 {\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 <a href="/facts/Lebesgue_integrable/f4AgvC6x">Lebesgue integrable</a>. 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 <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of (equivalence classes of) square integrable functions (with respect to <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>) forms the <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 space</a> with 
 
 
 
 p
 =
 2.
 
 
 {\displaystyle p=2.}
 
 Among the 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces, the class of square integrable functions is unique in being compatible with an <a href="/facts/Inner_product_space/HoyjElEy">inner product</a>, which allows notions like angle and orthogonality to be defined. Along with this inner product, the square integrable functions form a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>, since all of the 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
 spaces are <a href="/facts/Complete_metric_space/lsckmU8G">complete</a> under their respective <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 p
 
 
 {\displaystyle p}
 
-norms</a>.
Often the term is used not to refer to a specific function, but to equivalence classes of functions that are equal <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>.

Square-integrable function open-in-new

Square-integrable function