In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear^{[disambiguation needed]} functionals on L^p 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 convexity of the integral kernel. The result is attributed to the Italian mathematician Leonida Tonelli.

This article relies largely or entirely on a single source. (January 2013)

Let ${\displaystyle \Omega }$ be a bounded domain in ${\displaystyle n}$-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} \cup \{\pm \infty \}}$ be a continuous extended real-valued function. Define a nonlinear functional ${\displaystyle F}$ on functions ${\displaystyle u:\Omega \to \mathbb {R} ^{m}}$by

$${\displaystyle F[u]=\int _{\Omega }f(u(x))\,\mathrm {d} x.}$$

Then ${\displaystyle F}$ is sequentially weakly lower semicontinuous on the ${\displaystyle L^{p}}$ space ${\displaystyle L^{p}(\Omega ,\mathbb {R} ^{m})}$ for ${\displaystyle 1<p<+\infty }$ and weakly-∗ lower semicontinuous on ${\displaystyle L^{\infty }(\Omega ,\mathbb {R} ^{m})}$ if and only if ${\displaystyle f}$ is convex.

• Discontinuous linear functional