Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(Ga)$

We give a brief survey of the results on coarse or uniform embeddings of Banach spaces into $c_0(Ga)$ and the point character of Banach spaces. In the process we prove several new results in this direction (for example we determine the point character of the spaces $L_p(mu)$, $1le ple2$) solving open problems posed by C.~Avart, P.~Komjath, and V.~Roedl and by G.~Godefroy, G.~Lancien, and V.~Zizler. In particular, we show that $X=L_p(mu)$, $1le p<infty$, bi-Lipschitz embeds into $c_0(Ga)$ if and only if $dens X<om_om$.

我们简要综述了巴拿赫空间到\(c_0(\Gamma)\)的粗嵌入或一致嵌入以及巴拿赫空间的点特征方面的结果。在此过程中，我们在这个方向上证明了几个新的结果（例如，我们确定了空间\(L_p(\mu)\)，\(1\leq p\leq2\)的点特征），解决了C. 阿瓦特、P. 科姆亚什和V. 勒德尔以及G. 戈德弗鲁瓦、G. 朗西恩和V. 齐兹勒提出的公开问题。特别地，我们表明\(X = L_p(\mu)\)，\(1\leq p<\infty\)，双李普希茨嵌入到\(c_0(\Gamma)\)当且仅当\(dens X<\omega_{\omega}\)。