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In [[mathematics]], a '''square-integrable function''', also called a '''quadratically integrable function''', is a [[real number|real]]- or [[complex number|complex]]-valued [[measurable function]] for which the [[integral]] of the square of the [[absolute value]] is finite. Thus, if
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: <math> \int_{-\infty}^\infty |f(x)|^2 \, dx < \infty, </math>
 
then ''ƒ'' is square integrable on the real line <math>(-\infty,+\infty)</math>. One may also speak of quadratic integrability over bounded intervals such as&nbsp;[0,&nbsp;1].<ref>{{cite book
| title = Orthogonal Functions
| author = G.Sansone
| publisher = Dover Publications
| year = 1991
| isbn = 978-0-486-66730-0
| page = 1-2
}}</ref>
 
== Properties ==
The square integrable functions form an [[inner product space]] with [[inner product]] given by
 
: <math> \langle f, g \rangle = \int_A \overline{f(x)}g(x)\, dx </math>
 
where
* ''f'' and ''g'' are square integrable functions,
* {{overbar|''f''(''x'')}} is the [[complex conjugate]] of ''f'',
* ''A'' is the set over which one integrates&mdash;in the first example above,{{where|date=January 2014}} ''A'' is <math>(-\infty,+\infty)</math>; in the second, ''A'' is&nbsp;[0,&nbsp;1].
 
Since |''a''|<sup>2</sup>&nbsp;=&nbsp;''a&nbsp;{{overbar|a}}'', square integrability is the same as saying
 
: <math> \langle f, f \rangle < \infty. \, </math>
 
It can be shown<!-- cite theorem here, with an internal link?--> that square integrable functions form a [[complete metric space]] under the metric induced by the inner product defined above.
A complete metric space is also called a [[Cauchy space]], because sequences in such metric spaces converge if and only if they are [[Cauchy sequence|Cauchy]].
A space which is complete under the metric induced by a norm is a [[Banach 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 a [[Hilbert space]], because the space is complete under the metric induced by the inner product.
 
This inner product space is conventionally denoted by <math>\left(L_2, \langle\cdot, \cdot\rangle_2\right)</math> and many times abbreviated as <math>L_2</math>.
Note that <math>L_2</math> 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 <math>\langle\cdot, \cdot\rangle_2</math> specify the inner product space.
 
The space of square integrable functions is the [[Lp space|''L''<sup>''p''</sup> space]] in which&nbsp;''p''&nbsp;=&nbsp;2.
 
== References ==
{{Reflist}}
 
{{DEFAULTSORT:Quadratically Integrable Function}}
[[Category:Functional analysis]]

Latest revision as of 17:10, 7 January 2015

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