Rademacher's theorem
   HOME

TheInfoList



OR:

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, Rademacher's theorem, named after
Hans Rademacher Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German-born American mathematician, known for work in mathematical analysis and number theory. Biography Rademacher r ...
, states the following: If is an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of and is
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
, then is differentiable
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
in ; that is, the points in at which is ''not'' differentiable form a set of
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
zero. Differentiability here refers to infinitesimal approximability by a linear map, which in particular asserts the existence of the coordinate-wise partial derivatives.


Sketch of proof

The one-dimensional case of Rademacher's theorem is a standard result in introductory texts on measure-theoretic analysis. In this context, it is natural to prove the more general statement that any single-variable function of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' 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 conti ...
is differentiable almost everywhere. (This one-dimensional generalization of Rademacher's theorem fails to extend to higher dimensions.) One of the standard proofs of the general Rademacher theorem was found by Charles Morrey. In the following, let denote a Lipschitz-continuous function on . The first step of the proof is to show that, for any fixed unit vector , the -directional derivative of exists almost everywhere. This is a consequence of a special case of the
Fubini theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the ...
: a measurable set in has Lebesgue measure zero if its restriction to every line parallel to has (one-dimensional) Lebesgue measure zero. Considering in particular the set in where the -directional derivative of fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem. The second step of Morrey's proof establishes the linear dependence of the -directional derivative of upon . This is based upon the following identity: :\int_\frac\zeta(z)\,d\mathcal^n(x)=-\int_\fracf(x)\,d\mathcal^n(x). Using the Lipschitz assumption on , the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
can be applied to replace the two
difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact t ...
s in the above expression by the corresponding -directional derivatives. Then, based upon the known linear dependence of the -directional derivative of upon , the same can be proved of via the
fundamental lemma of calculus of variations In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero ...
. At this point in the proof, the existence of the gradient (defined as the -tuple of partial derivatives) is guaranteed to exist almost everywhere; for each , the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
with equals the -directional derivative almost everywhere (although perhaps on a smaller set). Hence, for any countable collection of unit vectors , there is a single set of measure zero such that the gradient and each -directional derivative exist everywhere on the complement of , and are linked by the dot product. By selecting to be dense in the unit sphere, it is possible to use the Lipschitz condition to prove the existence of ''every'' directional derivative everywhere on the complement of , together with its representation as the dot product of the gradient with the direction. Morrey's proof can also be put into the context of
generalized derivative In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet ...
s. Another proof, also via a reduction to the one-dimensional case, uses the technology of approximate limits.


Applications

Rademacher's theorem can be used to prove that, for any , the
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
is preserved under a bi-Lipschitz transformation of the domain, with the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
holding in its standard form. With appropriate modification, this also extends to the more general Sobolev spaces . Rademacher's theorem is also significant in the study of
geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
and
rectifiable set In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wi ...
s, as it allows the analysis of first-order differential geometry, specifically
tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s and
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
s. Higher-order concepts such as
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
remain more subtle, since their usual definitions require more differentiability than is achieved by the Rademacher theorem. In the presence of
convexity Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
, second-order differentiability is achieved by the
Alexandrov theorem In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if is an open subset of \R^n and f\colon U\to \R^m is a convex function, then f has a second derivative almost everywhere. In this conte ...
, the proof of which can be modeled on that of the Rademacher theorem. In some special cases, the Rademacher theorem is even used as part of the proof.


Generalizations

Alberto Calderón Alberto Pedro Calderón (September 14, 1920 – April 16, 1998) was an Argentinian mathematician. His name is associated with the University of Buenos Aires, but first and foremost with the University of Chicago, where Calderón and his mentor, t ...
proved the more general fact that if is an open bounded set in then every function in the
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
is differentiable almost everywhere, provided that . Calderón's theorem is a relatively direct corollary of the
Lebesgue differentiation theorem In mathematics, the Lebesgue differentiation theorem is a theorem of real analysis, which states that for almost every point, the value of an integrable function is the limit of infinitesimal averages taken about the point. The theorem is named for ...
and
Sobolev embedding theorem In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...
. Rademacher's theorem is a special case, due to the fact that any Lipschitz function on is an element of the space . There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
in terms of
metric differential In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can gene ...
s instead of the usual derivative.


See also

*
Pansu derivative In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by . A Carnot group G admits a one-parameter family of dilations, \delta_s\colon G\to G. If G_1 and G_2 are Carnot groups, then the Pansu derivative of a function f ...


References

Sources * * * * * * * * * *


External Links

* ''(Rademacher's theorem with a proof is on page 18 and further.)'' {{Reflist Lipschitz maps Theorems in measure theory