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Zbl 0954.46014
Bates, S.; Johnson, W.B.; Lindenstrauss, J.; Preiss, D.; Schechtman, G.
Affine approximation of Lipschitz functions and nonlinear quotients.
(English)
[J] Geom. Funct. Anal. 9, No.6, 1092-1127 (1999). ISSN 1016-443X; ISSN 1420-8970/e

For two Banach spaces $X$ and $Y$, $\text{Lip}(X,Y)$ is the set of Lipschitz functions from a domain in $X$ to $Y$ and $\text{Lip}(X,Y)$ has the approximation by affine property if for every Lipschitz function $f$ from the unit ball $B$ of $X$ into $Y$ and every $\varepsilon>0$ there is a ball $B_1\subset B$ of radius $r$ and an affine function $L\:X\to Y$ so that $\|f(x)-Lx\|\le\varepsilon r$, $x\in B_1$. Now $\text{Lip}(X,Y)$ has the uniform approximation by affine property if the radius $r=r(\varepsilon,f)$ of $B_1$ above can be chosen to satisfy $r(\varepsilon,f)\ge c(\varepsilon)>0$ simultaneously for all functions $f$ of Lipschitz constant $\le 1$. \par In section 2 of this paper several theorems are proved when such approximations are possible. Further, a complete characterization of the spaces $X$, $Y$ for which any Lipschitz function from $X$ to $Y$ can be so approximated is obtained in the following theorem: Let $X$ and $Y$ be nonzero Banach spaces. Then $\text{Lip}(X,Y)$ has the uniform approximation by affine property if and only if one of the spaces is super-reflexive and the other is finite dimensional. A uniformly continuous mapping $F$ from a metric space $X$ onto a metric space $Y$ is a uniform quotient mapping if for each $\varepsilon>0$ there is a $\delta=\delta(\varepsilon)>0$ so that for every $x\in X$, $F(B_\varepsilon(x))\supset B_\delta(Fx)$. A space $Y$ is said to be a uniform quotient of a space $X$ provided there is a uniform quotient mapping from $X$ onto $Y$. In section 3 the main theorem is the following: If $X$ is super-reflexive and $Y$ is a uniform quotient of~$X$ then $Y^*$ is finitely crudely representable in $X^*$. Consequently, $Y$ is isomorphic to a linear quotient of some ultra power of $X$. As a corollary the main theorem implies: If $Y$ is a uniform quotient of $L_p$, $1<p<\infty$, then $Y$ is isomorphic to a linear quotient of $L_p$.
MSC 2000:
*46E15 Banach spaces of functions defined by smoothness properties
41A10 Approximation by polynomials

Keywords: Lipschitz function; (uniform) approximation by affine property; uniform quotient mapping; superreflexive; finitely crudely representable; ultra power

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