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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