Saturday, 23 June 2007

ca.analysis and odes - Lipschitz functions in mathbbRnmathbbRn

Let f=(f1,ldots,fn):[a,b]rightarrowmathbbRnf=(f1,ldots,fn):[a,b]rightarrowmathbbRn be a continuously differentiable function. (See the comments above for an explanation as to why the hypotheses have been strengthened.)



For 1leqileqn1leqileqn, let



Li=maxxin[a,b]|fi(x)|Li=maxxin[a,b]|fi(x)|,



so that, by the Mean Value Theorem, for x,yin[a,b]x,yin[a,b],



|fi(x)fi(y)|=|fi(c)||xy|leqLi|xy||fi(x)fi(y)|=|fi(c)||xy|leqLi|xy|.



Then, taking the standard Euclidean norm on mathbbRnmathbbRn,



|f(x)f(y)|2=sumni=1|fi(x)fi(y)|2leq(sumni=1L2i)|xy|2|f(x)f(y)|2=sumni=1|fi(x)fi(y)|2leq(sumni=1L2i)|xy|2,



so



|f(x)f(y)|leqsqrt(sumni=1L2i)|xy||f(x)f(y)|leqsqrt(sumni=1L2i)|xy|.



Thus we can take



L=sqrtsumni=1L2iL=sqrtsumni=1L2i.



Since all norms on mathbbRnmathbbRn are equivalent -- i.e., differ at most by a multiplicative constant -- the choice of norm on mathbbRnmathbbRn will change the expression of the Lipschitz constant LL in terms of the Lipschitz constants LiLi of the components, but not whether f is Lipschitz.

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