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Lipschitz spaces and M-ideals

Heiko Berninger·Dirk Werner
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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 $\ell^{\infty}$.

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Heiko Berninger, Dirk Werner. 2002-01-16. Lipschitz spaces and M-ideals. https://arxiv.org/abs/math/0201144

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