Functions

With appropriate adjustment of the value of the Lipschitz con­ stant c the norm in (1) can be any norm on Rn since all such norms are equivalent. That is, given norms L, M on Rn there exists c > 0 such that L(u) < cM (u) for all u in Rn. The familiar norms on Rn are of the form

(2)

for u = (tti, · · · , un) where the parameter p satisfies 1 < p < oo. For p = oo we have the norm

The Euclidean norm is given by p = 2 in (2). For convenience we often use (2) with p = 1 since it has the simple expression

For the space D of all differentials on a cell K each Lipschitz function F on Rn induces a mapping F of Dn into D defined by

(5)

F (S i , · · · ,S n) at ( / , t) is F (S i(I ,t) , · · · ,S n (I,t)). To show that (5) is an effective definition we must verify that

(6) To do so set u = (Si, · · · , Sn) and v = (SJ, · · · , S'n) in (1) with the norm (4) to get the summant inequality

which immediately gives (6). A norm L on Rn is a Lipschitz function with Lipschitz con­

stant c = 1 by the triangle inequality. So L defines a differen­ tial L(a i, · · · ,σ η) on K for σχ, · · · ,σ η differentials on K . For L(u) = ||u ||p in (2) or (3) we shall use the notation

for 1 < p < oo and

(8) In general we may use “F ” in place of “F ” in (5). This is convenient. But care must be taken since a Lipschitz function may be composed of functions some of which are not Lipschitzean. For example if for i = 1, · · · , n cq is continuous and weakly archimedean then af — 0. So one might draw the false conclusion that

But the square root \τ\χ/ 2 is not defined for differentials r since F(u) = \u\1/2 fails to be Lipschitzean on M near 0.