For a Banach space X its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and Z is linearly dense in X for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced recently by T. Russo and J. Somaglia but overcomplete sets have been considered in separable Banach spaces since the 1950ties.

We prove some absolute and consistency results concerning the existence and the nonexistence of overcomplete sets in some classical nonseparable Banach spaces. For example: $c_0({\omega }_1)$, $C([0,{\omega }_1])$, $L_1({\{0,1\}}^{{\omega }_1})$, ${\ell }_p({\omega }_1)$, $L_p({\{0,1\}}^{{\omega }_1})$ for $p\in (1,\infty )$ or in general WLD Banach spaces of density ${\omega }_1$ admit overcomplete sets (in ZFC). The spaces ${\ell }_{\infty }$, ${\ell }_{\infty }/c_0$, spaces of the form $C(K)$ for K extremally disconnected, superspaces of ${\ell }_1({\omega }_1)$ of density ${\omega }_1$ do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generated in ${\ell }_{\infty }$ by $c_0$ and the characteristic functions of elements of an almost disjoint family of subsets of $\mathbb{N}$ of cardinality ${\omega }_1$ admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density ${\omega }_1$ which are separable in the weak^⁎ topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open.

对于 Banach 空间X其子集$是\subseteq X$ 如果 $|是|=d\mathrm{电子}n秒(X)$并且Z在X 中线性稠密，对于每个$Z\subseteq 是$ 和 $|Z|=|是|$. 在不可分离的 Banach 空间的上下文中，这个概念最近由 T. Russo 和 J. Somaglia 引入，但自 1950 年代以来，在可分离的 Banach 空间中已经考虑过超完备集。

我们证明了一些经典不可分 Banach 空间中过完备集存在与不存在的绝对一致性结果。例如：$C_0({\omega }_1)$, $C([0,{\omega }_1])$, ${升}_1({\{0,1\}}^{{\omega }_1})$, ${\ell }_{磷}({\omega }_1)$, ${升}_{磷}({\{0,1\}}^{{\omega }_1})$ 为了 $磷\in (1,\infty )$ 或一般而言，密度的 WLD Banach 空间 ${\omega }_1$承认过完备集（在 ZFC 中）。空间${\ell }_{\infty }$, ${\ell }_{\infty }/C_0$, 形式的空格 $C(钾)$对于K极不连通的超空间${\ell }_1({\omega }_1)$ 密度 ${\omega }_1$不承认过完备集（在 ZFC 中）。Johnson-Lindenstrauss 空间是否在${\ell }_{\infty }$ 经过 $C_0$ 以及几乎不相交的子集族的元素的特征函数 $\mathbb{N}$ 基数的 ${\omega }_1$承认一个超完备集是不可判定的。这同样适用于所有具有双密度球的不可分巴拿赫空间${\omega }_1$它们在弱^⁎拓扑中是可分离的。结果证明涉及更广泛的 Banach 空间类别，但几个自然开放的问题仍然悬而未决。