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We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we cannot achieve a satisfactory extension of Rosenthal's ℓ₁-theorem to spaces of the type ℓ₁(κ) for κ an uncountable cardinal.

Suppose that $X$ is a Fréchet space, $\langle a_{ij}\rangle$ is a regular method of summability and $\left(x_i\right)$ is a bounded sequence in $X$. We prove that there exists a subsequence $\left(y_i\right)$ of $\left(x_i\right)$ such that: either (a) all the subsequences of $\left(y_i\right)$ are summable to a common limit with respect to $\langle a_{ij}\rangle$; or (b) no subsequence of $\left(y_i\right)$ is summable with respect to $\langle a_{ij}\rangle$. This result generalizes the Erdös-Magidor theorem which refers to summability of bounded sequences in Banach spaces. We also show that two analogous results for some ${\omega }_1$-locally convex spaces...

We consider a Banach space, which comes naturally from $c_0$ and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings defined on weakly compact, convex sets.