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2017 The $\ell_1$-Dichotomy Theorem with Respect to a Coideal
Vassiliki Farmaki, Andreas Mitropoulos
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Real Anal. Exchange 42(1): 167-184 (2017).

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 $\ell_1(B)$.

Citation

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Vassiliki Farmaki. Andreas Mitropoulos. "The $\ell_1$-Dichotomy Theorem with Respect to a Coideal." Real Anal. Exchange 42 (1) 167 - 184, 2017.

Information

Published: 2017
First available in Project Euclid: 27 March 2017

zbMATH: 06848945
MathSciNet: MR3702560

Subjects:
Primary: 54A20, ‎54C30
Secondary: ‎54C30

Rights: Copyright © 2017 Michigan State University Press

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Vol.42 • No. 1 • 2017
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