The $\ell_1$-Dichotomy Theorem with Respect to a Coideal

Abstract

In this paper we introduce, for any coideal basis $\mathcal{B}$ on the set $\nat$ of natural numbers, the notions of a $\mathcal{B}$-sequence, a $\mathcal{B}$-subsequence of a $\mathcal{B}$-sequence, and a $\mathcal{B}$-convergent sequence in a metric space. The usual notions of a sequence, subsequence, and convergent sequence obtain for the coideal $\mathcal{B}$ of all the infinite subsets of $\nat$. We first prove a Bolzano-Weierstrass theorem for $\mathcal{B}$-sequences: if $\mathcal{B}$ is a Ramsey coideal basis on $\nat$, then every bounded $\mathcal{B}$-sequence of real numbers has a $\mathcal{B}$-convergent $\mathcal{B}$-subsequence; and next, with the help of this extended Bolzano-Weierstrass theorem, we establish an extension of the fundamental Rosenthal's $\ell_1$-dichotomy theorem: if $\mathcal{B} $ is a semiselective coideal basis on $\mathbb{N}$, then every bounded $\mathcal{B}$-sequence of real valued functions $(f_n)_{n\in A}$ has a $\mathcal{B}$-subsequence $(f_n)_{n\in B}$, which is either $\mathcal{B}$-convergent or equivalent to the unit vector basis of $\ell_1(B)$.