Mathematics – Functional Analysis
Scientific paper
2010-05-19
Mathematics
Functional Analysis
Scientific paper
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 $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where $\mu_\xi$'s are atomic measures with supports consisting of at most $2n-1$ points of $K_{2n}$, but there are biorthogonal systems $(f_\xi,\mu_\xi)_{\xi\in \omega_1}$ where $\mu_\xi$'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 $\delta_{x_\xi}-\delta_{y_\xi}$ for $\xi<\omega_1$ and $x_\xi,y_\xi\in K$. 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 $k>n$.
Brech Christina
Koszmider Piotr
No associations
LandOfFree
On biorthogonal systems whose functionals are finitely supported does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On biorthogonal systems whose functionals are finitely supported, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On biorthogonal systems whose functionals are finitely supported will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-480390