Rademacher's theorem

In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rⁿ and

f : U → R^m

is Lipschitz continuous, then f is Fréchet-differentiable almost everywhere in U (i.e. the points in U at which f is not differentiable form a set of Lebesgue measure zero).

Generalizations

There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.

References

• Juha Heinonen, Lectures on Lipschitz Analysis, Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.)

This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.v · Categories: Mathematical analysis stubs Lipschitz maps Theorems in measure theory Wikimedia Foundation. 2010. Игры ⚽ Поможем написать реферат Real Security Act of 2006 Burdette Park Look at other dictionaries: Hans Rademacher — Hans Adolph Rademacher (3 April 1892, Wandsbeck, now Hamburg Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German mathematician, known for work in mathematical analysis and number theory. He emigrated from Europe in 1934.… …   Wikipedia Midy's theorem — In mathematics, Midy s theorem, named after French mathematician E. Midy,[1] is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period. If the period of the… …   Wikipedia Jung's theorem — In geometry, Jung s theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901.… …   Wikipedia Scientific phenomena named after people — This is a list of scientific phenomena and concepts named after people (eponymous phenomena). For other lists of eponyms, see eponym. NOTOC A* Abderhalden ninhydrin reaction Emil Abderhalden * Abney effect, Abney s law of additivity William de… …   Wikipedia List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… …   Wikipedia 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… …   Wikipedia Integration by substitution — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia Lipschitz continuity — In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity. Intuitively, a Lipschitz continuous function is limited in… …   Wikipedia Hardy-Littlewood maximal function — In mathematics, the Hardy Littlewood maximal operator M is a significant non linear operator used in real analysis and harmonic analysis. It takes a function f (a complex valued and locally integrable function) : f:mathbb{R}^{d} ightarrow… …   Wikipedia Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”