Lipschitz condition and differentiability

If X and Y are Banach spacesMathworldPlanetmath, e.g. n, one can inquire about the relationMathworldPlanetmath between differentiability and the Lipschitz conditionMathworldPlanetmath. If f is Lipschitz, the ratio


is boundedPlanetmathPlanetmathPlanetmathPlanetmath but is not assumed to convergePlanetmathPlanetmath to a limit.

Proposition 1

Let f:XY be a continuously differentiable mapping ( between Banach spaces. If KX is a compact subset, then the restrictionPlanetmathPlanetmathPlanetmath f:KY satisfies the Lipschitz condition.

Proof. Let lin(X,Y) denote the Banach space of bounded linear maps from X to Y. Recall that the norm T of a linear mapping Tlin(X,Y) is defined by


Let Df:Xlin(X,Y) denote the derivativePlanetmathPlanetmath of f. By definition Df is continuousMathworldPlanetmathPlanetmath, which really means that Df:X is a continuous function. Since KX is compact, there exists a finite upper bound B1>0 for Df restricted to K. In particular, this means that


for all pK,uX.

Next, consider the secant mapping s:X×X defined by


This mapping is continuous, because f is assumed to be continuously differentiable. Hence, there is a finite upper bound B2>0 for s restricted to the compact set K×K. It follows that for all p,qK we have

f(q)-f(p) f(q)-f(p)-Df(p)(q-p)+Df(p)(q-p)

Therefore B1+B2 is the desired Lipschitz constant. QED

Neither condition is stronger. For example, the function f: given by f(x)=x2 is differentiableMathworldPlanetmath but not Lipschitz.

Title Lipschitz condition and differentiability
Canonical name LipschitzConditionAndDifferentiability
Date of creation 2013-03-22 11:57:50
Last modified on 2013-03-22 11:57:50
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 34
Author Mathprof (13753)
Entry type Theorem
Classification msc 26A16
Synonym mean value inequality
Related topic Derivative2