Editing Lp space (section)

===Dual spaces===

The [[Continuous dual|dual space]] of <math>L^p(\mu)</math> for <math>1 < p < \infty</math> has a natural isomorphism with <math>L^q(\mu),</math> where <math>q</math> is such that <math>\tfrac{1}{p} + \tfrac{1}{q} = 1</math>. This isomorphism associates <math>g \in L^q(\mu)</math> with the functional <math>\kappa_p(g) \in L^p(\mu)^*</math> defined by
<math display="block">f \mapsto \kappa_p(g)(f) = \int f g \, \mathrm{d}\mu</math> 
for every <math>f \in L^p(\mu).</math>

<math>\kappa_p : L^q(\mu) \to L^p(\mu)^*</math> is a well defined continuous linear mapping which is an [[isometry]] by the [[Hölder's inequality#Extremal equality|extremal case]] of Hölder's inequality. If <math>(S,\Sigma,\mu)</math> is a [[Measure_space#Important_classes_of_measure_spaces|<math>\sigma</math>-finite measure space]] one can use the [[Radon–Nikodym theorem]] to show that any <math>G \in L^p(\mu)^*</math> can be expressed this way, i.e., <math>\kappa_p</math> is an [[Isometry#Definition|isometric isomorphism]] of [[Banach space]]s.{{sfn|Rudin|1987|loc=Theorem 6.16}} Hence, it is usual to say simply that <math>L^q(\mu)</math> is the [[continuous dual space]] of <math>L^p(\mu).</math>

For <math>1 < p < \infty,</math> the space <math>L^p(\mu)</math> is [[reflexive space|reflexive]]. Let <math>\kappa_p</math> be as above and let <math>\kappa_q : L^p(\mu) \to L^q(\mu)^*</math> be the corresponding linear isometry. Consider the map from <math>L^p(\mu)</math> to <math>L^p(\mu)^{**},</math> obtained by composing <math>\kappa_q</math> with the [[dual space#Transpose of a continuous linear map|transpose]] (or adjoint) of the inverse of <math>\kappa_p:</math>

<math display="block">j_p : L^p(\mu) \mathrel{\overset{\kappa_q}{\longrightarrow}} L^q(\mu)^* \mathrel{\overset{\left(\kappa_p^{-1}\right)^*}{\longrightarrow}} L^p(\mu)^{**}</math>

This map coincides with the [[Reflexive space#Definitions|canonical embedding]] <math>J</math> of <math>L^p(\mu)</math> into its bidual. Moreover, the map <math>j_p</math> is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure <math>\mu</math> on <math>S</math> is [[sigma-finite]], then the dual of <math>L^1(\mu)</math> is isometrically isomorphic to <math>L^\infty(\mu)</math> (more precisely, the map <math>\kappa_1</math> corresponding to <math>p = 1</math> is an isometry from <math>L^\infty(\mu)</math> onto <math>L^1(\mu)^*.</math>

The dual of <math>L^\infty(\mu)</math> is subtler. Elements of <math>L^\infty(\mu)^*</math> can be identified with bounded signed ''finitely'' additive measures on <math>S</math> that are [[absolutely continuous]] with respect to <math>\mu.</math> See [[ba space]] for more details. If we assume the axiom of choice, this space is much bigger than <math>L^1(\mu)</math> except in some trivial cases. However, [[Saharon Shelah]] proved that there are relatively consistent extensions of [[Zermelo–Fraenkel set theory]] (ZF + [[Axiom of dependent choice|DC]] + "Every subset of the real numbers has the [[Baire property]]") in which the dual of <math>\ell^\infty</math> is <math>\ell^1.</math><ref>{{Citation|title=Handbook of Analysis and its Foundations|last=Schechter |first=Eric|year=1997| publisher=Academic Press Inc.|location=London}} See Sections 14.77 and 27.44–47</ref>