论文标题
关于在某些经典不可分割的Banach空间中存在过度集合的集合
On the existence of overcomplete sets in some classical nonseparable Banach spaces
论文作者
论文摘要
对于Banach Space $ x $，如果$ | y | = dens（x）$，而$ z $的子集$ y \ y \ subseteq x $被称为userplete，则在$ x $中是$ x $ in $ z \ subseteq y $，in $ x $ in $ x $，with $ | z | z | | | = | y | $。在不可分割的Banach空间的背景下，这一概念是T. Russo和J. Somaglia最近引入的，但是自1950年代以来，在可分开的Banach空间中考虑了过度的集合。 我们证明了有关某些经典不可分割的Banach空间中过度集合的存在和不存在的绝对和一致性结果。例如：$ c_0（ω_1）$，$ c（[0，ω_1]）$，$ l_1（\ {0,1 \}^{ω_1}）$，$ \ ell_p（ω_1）$，$ l_p（$ l_p）密度$ω_1$的Banach空间允许套装集（以ZFC为单位）。空间$ \ ell_ \ infty $，$ \ ell_ \ infty/c_0 $，$ c（k）$的空格，$ k $非常断开连接，超级空间$ \ ell_1（ω_1）$密度$ω_1$的$ \ ell_1（ω_1）$不允许completecomplete completeplete sets（在ZFC中）。 Johnson-Lindenstrauss空间是否生成$ \ ell_ \ by $ C_0 $以及几乎不连接的$ \ Mathbb n $ cardinality $ω_1$的子集的元素的特征函数，承认超出的集合是不可确定的。相同的是所有不可分割的Banach空间，其密度$ω_1$的双球在弱$^*$拓扑中是可分离的。结果证明是指更广泛的Banach空间，但几个自然的开放问题仍然开放。
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: $c_0(ω_1)$, $C([0,ω_1])$, $L_1(\{0,1\}^{ω_1})$, $\ell_p(ω_1)$, $L_p(\{0,1\}^{ω_1})$ for $p\in (1, \infty)$ or in general WLD Banach spaces of density $ω_1$ 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 $\ell_1(ω_1)$ of density $ω_1$ do not admit overcomplete sets (in ZFC). Whether the Johnson-Lindenstrauss space generatedin $\ell_\infty$ by $c_0$ and the characteristic functions of elements of an almost disjoint family of subsets of $\mathbb N$ of cardinality $ω_1$ admits an overcomplete set is undecidable. The same refers to all nonseparable Banach spaces with the dual balls of density $ω_1$ which are separable in the weak$^*$ topology. The results proved refer to wider classes of Banach spaces but several natural open questions remain open.
