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-In [[mathematics]], the '''Paley–Wiener integral''' is a simple [[stochastic integral]]. When applied to [[Abstract Wiener space|classical Wiener space]], it is less general than the [[Itō integral]], but the two agree when they are both defined.
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-The integral is named after its discoverers, [[Raymond Paley]] and [[Norbert Wiener]].
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-Let ''i''&nbsp;:&nbsp;''H''&nbsp;→&nbsp;''E'' be an [[abstract Wiener space]] with abstract Wiener measure ''γ'' on ''E''. Let ''j''&nbsp;:&nbsp;''E''<sup>∗</sup>&nbsp;→&nbsp;''H'' be the [[adjoint of an operator|adjoint]] of ''i''. (We have abused notation slightly: strictly speaking, ''j''&nbsp;:&nbsp;''E''<sup>∗</sup>&nbsp;→&nbsp;''H''<sup>∗</sup>, but since ''H'' is a [[Hilbert space]], it is [[Isometry|isometrically isomorphic]] to its [[dual space]] ''H''<sup>∗</sup>, by the [[Riesz representation theorem]].{{Citation needed|date=September 2010}})
-It can be shown that ''j'' is an [[injective function]] and has [[dense (topology)|dense]] [[image (function)|image]] in ''H''.{{Citation needed|date=September 2010}} Furthermore, it can be shown that every [[linear functional]] ''f''&nbsp;∈&nbsp;''E''<sup>∗</sup> is also [[square-integrable]]: in fact,
-:<math>\| f \|_{L^{2} (E, \gamma; \mathbb{R})} = \| j(f) \|_{H}</math>
-This defines a natural [[linear map]] from ''j''(''E''<sup>∗</sup>) to ''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R'''), under which ''j''(''f'')&nbsp;∈&nbsp;''j''(''E''<sup>∗</sup>)&nbsp;⊆&nbsp;''H'' goes to the [[equivalence class]] [''f''] of ''f'' in ''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R'''). This is well-defined since ''j'' is injective. This map is an [[isometry]], so it is [[continuous function|continuous]].
-However, since a continuous linear map between [[Banach space]]s such as ''H'' and ''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R''') is uniquely determined by its values on any dense subspace of its domain, there is a unique continuous linear extension ''I''&nbsp;:&nbsp;''H''&nbsp;→&nbsp;''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R''') of the above natural map ''j''(''E''<sup>∗</sup>)&nbsp;→&nbsp;''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R''') to the whole of ''H''.
-This isometry ''I''&nbsp;:&nbsp;''H''&nbsp;→&nbsp;''L''<sup>2</sup>(''E'',&nbsp;''γ'';&nbsp;'''R''') is known as the '''Paley–Wiener map'''. ''I''(''h''), also denoted &lt;''h'',&nbsp;&minus;&gt;<sup>∼</sup>, is a function on ''E'' and is known as the '''Paley–Wiener integral''' (with respect to ''h''&nbsp;∈&nbsp;''H'').
-It is important to note that the Paley–Wiener integral for a particular element ''h''&nbsp;∈&nbsp;''H'' is a [[Function (mathematics)|function]] on ''E''. The notation &lt;''h'',&nbsp;''x''&gt;<sup>∼</sup> does not really denote an inner product (since ''h'' and ''x'' belong to two different spaces), but is a convenient [[abuse of notation]] in view of the [[Cameron–Martin theorem]]. For this reason, many authors{{Citation needed|date=September 2010}} prefer to write &lt;''h'',&nbsp;&minus;&gt;<sup>∼</sup>(''x'') or ''I''(''h'')(''x'') rather than using the more compact but potentially confusing &lt;''h'',&nbsp;''x''&gt;<sup>∼</sup> notation.
-==See also==
-Other stochastic integrals:
-* [[Itō integral]]
-* [[Skorokhod integral]]
-* [[Stratonovich integral]]
-{{No footnotes|date=September 2010}}
-==References==
-*Park, C.; Skoug, D. (1988) "A Note on Paley-Wiener-Zygmund Stochastic Integrals", ''Proceedings of the American Mathematical Society', 103 (2), 591&ndash;601 {{JSTOR|2047184}}
-*Elworthy, D. (2008) [http://www.tjsullivan.org.uk/pdf/MA482_Stochastic_Analysis.pdf ''MA482 Stochastic Analysis''],  Lecture Notes, University of Warwick (Section 6)
-{{DEFAULTSORT:Paley-Wiener Integral}}
-[[Category:Definitions of mathematical integration]]
-[[Category:Stochastic calculus]]

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