Preprint No.
A-02-02
Heiko Berninger,
Dirk Werner
Lipschitz spaces and M-ideals
Abstract:
For a metric space $(K,d)$ the Banach space $\Lip(K)$ consists of all
scalar-valued bounded Lipschitz functions on $K$ with the norm
$\|f\|_{L}=\max(\|f\|_{\infty},L(f))$, where $L(f)$ is the Lipschitz
constant of $f$. The closed subspace $\lip(K)$ of $\Lip(K)$ contains
all elements of $\Lip(K)$ satisfying the $\lip$-condition
$\lim_{0 < d(x,y)\to 0}|f(x)-f(y)|/d(x,y)=0$. For
$K=([0,1],|\,{\cdot}\,|^{\alpha})$, $0 < \alpha < 1$, we prove that
$\lip(K)$ is a proper $M$-ideal in
a certain subspace of $\Lip(K)$ containing a copy of $\ell^{\infty}$.
Keywords: 3-ball property, Hölder spaces, Lipschitz spaces, little Lipschitz spaces, M-embedded spaces, M-ideals
Mathematics Subject Classification (MSC2000): 46B04, 46B20, 46E15
Language: ENG
Available: Pr-A-02-02.ps
Contact: Dirk Werner, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (werner@math.fu-berlin.de)
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