View source for Tonelli's theorem (functional analysis)

{{Other uses|Tonelli's theorem (disambiguation){{!}}Tonelli's theorem}}
{{one source|date=January 2013}}
In [[mathematics]], '''Tonelli's theorem in functional analysis''' is a fundamental result on the [[weak topology|weak]] [[lower semicontinuous|lower semicontinuity]] of [[nonlinear]] [[functional (mathematics)|functionals]] on [[Lp space|''L''<sup>''p''</sup> spaces]]. As such, it has major implications for [[functional analysis]] and the [[calculus of variations]]. Roughly, it shows that weak lower semicontinuity for integral functionals is equivalent to [[convex function|convexity]] of the integral kernel. The result is attributed to the [[Italy|Italian]] [[mathematician]] [[Leonida Tonelli]].

==Statement of the theorem==

Let <math>\Omega</math> be a bounded [[Domain (mathematical analysis)|domain]] in <math>n</math>-[[dimension]]al [[Euclidean space]] <math>\Reals^n</math> and let <math>f : \Reals^m \to \Reals \cup \{\pm \infty\}</math> be a [[Continuous function|continuous]] [[extended real number line|extended real-valued]] function. Define a nonlinear functional <math>F</math> on functions <math>u : \Omega \to \Reals^m</math>by
<math display=block>F[u] = \int_{\Omega} f(u(x)) \, \mathrm{d} x.</math>

Then <math>F</math> is sequentially weakly lower [[Semi-continuity|semicontinuous]] on the <math>L^p</math> space <math>L^p(\Omega, \Reals^m)</math> for <math>1 < p < +\infty</math> and weakly-&lowast; lower semicontinuous on <math>L^\infty(\Omega, \Reals^m)</math> [[if and only if]] <math>f</math> is [[Convex function|convex]].

==See also==

* {{annotated link|Discontinuous linear functional}}

==References==

{{reflist}}

* {{cite book
|author1=Renardy, Michael
|author2=Rogers, Robert C.
|name-list-style=amp
|title=An introduction to partial differential equations
|   series=Texts in Applied Mathematics 13
|  edition=Second
|publisher=Springer-Verlag
| location=New York
|     year=2004
|    pages=347
|     isbn=0-387-00444-0
}} (Theorem 10.16)

{{Convex analysis and variational analysis}}
{{Measure theory}}
{{Banach spaces}}

[[Category:Calculus of variations]]
[[Category:Convex analysis]]
[[Category:Function spaces]]
[[Category:Measure theory]]
[[Category:Theorems in functional analysis]]
[[Category:Variational analysis]]