3
votes
2answers
139 views

Operators on $\ell_\infty(\Gamma)$ and almost disjoint families of subsets

It is well known that given an operator $T:\ell_\infty\to\ell_\infty$ such that $Tx=0$ for each $x\in c_0$ there exists an infinite subset $M$ of the positive integers so that $Tx=0$ for each $x\in ...
3
votes
1answer
144 views

Algebras with countable chains only

Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
4
votes
1answer
218 views

Density character of $\ell_\infty(\kappa, S)$

Let $S$ be an uncountable set. Consider the subspace $\ell_\infty(\kappa, S)$ of $\ell_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). ...
2
votes
1answer
253 views

Uniqueness of dimension in Banach spaces

Let $\;\;\; \big\langle V,+,\cdot, \;\; ||\cdot|| \;\; \big\rangle \;\;\;$ be a Banach space over the field $\mathbb{K}$, which is either $\mathbb{R}$ or $\mathbb{C}$. Suppose there exists a subset ...
3
votes
3answers
384 views

Shape of long sequences in C(ω_1)

Apologies for the vague title - I couldn't come up with a single sentence that summarised this problem well. If you can, please edit or suggest a better one! This question is also rather specific and ...