泛函支集的双正交系
On biorthogonal systems whose functionals are finitely supported
摘要：
我们表明，对于每个自然$n>1$是一致的，有一个紧凑的Hausdorff空间$K_{2n}$，使在$C(K_{2n})$没有不可数的（半）双正交序列$(f_ξ,μ_ξ)_{ξ\in ω_1}$，其中$μ_ξ$的原子措施，支持组成的最$2n-1$点，但有双正交系$(f_ξ,μ_ξ)_{ξ\in ω_1}$，其中$μ_ξ$的原子措施，支持组成的$2n$点。这补充了Todorcevic的一个结果，即每个不可分的Banach空间$C(K)$都有一个不可数的双正交系统，其中泛函是$ξ<ω_1$和$x_ξ,y_ξ\in K$的$δ_{x_ξ}-δ_{y_ξ}$形式的测度。证明了对于$K$完全不连通的布尔代数$Clop(K)$或Banach代数$C(K)$的不可约性可以严格小于$C(K)$中双正交系的大小.紧空间表现出一个有趣的行为相对于已知的基数函数：遗传密度的权力$K_{2n}^k$是可数的$k=n$和它是不可数的（甚至蔓延是不可数的）$k>n$。
英文摘要：
We show that for each natural $n>1$ it is consistent that there is a compact Hausdorff space $K_{2n}$ such that in $C(K_{2n})$ there is no uncountable (semi)biorthogonal sequence $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of at most $2n-1$ points of $K_{2n}$, but there are biorthogonal systems $(f_ξ,μ_ξ)_{ξ\in ω_1}$ where $μ_ξ$'s are atomic measures with supports consisting of $2n$ points. This complements a result of Todorcevic that it is consistent that each nonseparable Banach space $C(K)$ has an uncountable biorthogonal system where the functionals are measures of the form $δ_{x_ξ}-δ_{y_ξ}$ for $ξ<ω_1$ and $x_ξ,y_ξ\in K$. It also follows that it is consistent that the irredundance of the Boolean algebra $Clop(K)$ or the Banach algebra $C(K)$ for $K$ totally disconnected can be strictly smaller than the sizes of biorthogonal systems in $C(K)$. The compact spaces exhibit an interesting behaviour with respect to known cardinal functions: the hereditary density of the powers $K_{2n}^k$ is countable up to $k=n$ and it is uncountable (even the spread is uncountable) for $k>n$.
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