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In mathematics, **Tonelli's theorem in functional analysis** is a fundamental result on the weak lower semicontinuity of nonlinear 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.

Let Ω be a bounded domain in *n*-dimensional Euclidean space **R**^{n} and let *f* : **R**^{m} → **R** ∪ {±∞} be a continuous extended real-valued function. Define a nonlinear functional *F* on functions *u* : Ω → **R**^{m} by

Then *F* is sequentially weakly lower semicontinuous on the *L*^{p} space *L*^{p}(Ω; **R**^{m}) for 1 < *p* < +∞ and weakly-∗ lower semicontinuous on *L*^{∞}(Ω; **R**^{m}) if and only if the function *f*

is convex.

In mathematical analysis, **semi-continuity** is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function *f* is **upper****semi-continuous** at a point *x*_{0} if, roughly speaking, the function values for arguments near *x*_{0} are not much higher than *f*(*x*_{0}).

In mathematical analysis, a function of **bounded variation**, also known as ** BV function**, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the

In mathematics, a **Radon measure**, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space *X* that is finite on all compact sets, outer regular on all Borel sets, and inner regular on open sets. These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures.

In mathematics and in signal processing, the **Hilbert transform** is a specific linear operator that takes a function, *u*(*t*) of a real variable and produces another function of a real variable *H*(*u*)(*t*). This linear operator is given by convolution with the function :

In mathematics, a **Sobolev space** is a vector space of functions equipped with a norm that is a combination of *L ^{p}*-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

In mathematics, the **total variation** identifies several slightly different concepts, related to the structure of the codomain of a function or a measure. For a real-valued continuous function *f*, defined on an interval [*a*, *b*] ⊂ ℝ, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation *x* ↦ *f*(*x*), for *x* ∈ [*a*, *b*].

In functional analysis, a branch of mathematics, the **Borel functional calculus** is a *functional calculus*, which has particularly broad scope. Thus for instance if *T* is an operator, applying the squaring function *s* → *s*^{2} to *T* yields the operator *T*^{2}. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

In mathematics, the **Gateaux differential** or **Gateaux derivative** is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, a French mathematician who died young in World War I, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.

In mathematics, **plurisubharmonic** functions form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions plurisubharmonic functions can be defined in full generality on complex analytic spaces.

In mathematics, **Tonelli's theorem** may refer to

In mathematical analysis, **Trudinger's theorem** or the **Trudinger inequality** is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger.

In mathematics, a **locally integrable function** is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to *L*^{p} spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain : in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.

In mathematics, **Friedrichs's inequality** is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the *L ^{p}* norm of a function using

In mathematics, a **Caccioppoli set** is a set whose boundary is measurable and has a *finite measure*. A synonym is **set of (locally) finite perimeter**. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation.

In mathematics, the **Hardy–Littlewood maximal operator***M* is a significant non-linear operator used in real analysis and harmonic analysis. It takes a locally integrable function *f* : **R**^{d} → **C** and returns another function *Mf* that, at each point *x* ∈ **R**^{d}, gives the maximum average value that *f* can have on balls centered at that point. More precisely,

In mathematics—specifically, in functional analysis—a **weakly measurable function** taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.

In mathematics, the **Pettis integral** or **Gelfand–Pettis integral**, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the **weak integral** in contrast to the Bochner integral, which is the strong integral.

In mathematical analysis, **Ekeland's variational principle**, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems.

In mathematics, the **direct method in the calculus of variations** is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.

The **maximum theorem** provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control.

- Renardy, Michael & Rogers, Robert C. (2004).
*An introduction to partial differential equations*. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 347. ISBN 0-387-00444-0. (Theorem 10.16)

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