\(\epsilon\)-weak Cauchy sequences and a quantitative version of Rosenthal's \(\ell_1\)-theorem
The author takes up and refines (especially for complex scalars) \textit{E. Behrends}'s quantified version [Bull. Pol. Acad. Sci., Math. 43, No. 4, 283--295 (1995; Zbl 0847.46007)] of Rosenthal's \(\ell^1\)-theorem. (Recall that the latter states that a bounded sequence in a (real or complex) Banach space admits a subsequence which is either weakly Cauchy or equivalent to \(\ell^1\)'s canonical basis.) The setting is a compact Hausdorff space \(K\), a bounded sequence \((f_n)\) of (real or complex) continuous functions in \(C(K)\) and the quantified notion of \((f_n)\) being \(\varepsilon\)-weakly Cauchy on \(K\) if, for every \(t\in K\), there exists \(n_0\in\mathbb{N}\) such that \(|f_n(t)-f_m(t)|<\varepsilon\) for all \(n,m\geq n_0\). (This natural notion is implicit in Behrends's paper and resembles the \(\delta\)-modulus in papers like [\textit{O. F. K. Kalenda} and \textit{J. Spurný}, Bull. Aust. Math. Soc. 91, No. 3, 471--486 (2015; Zbl 1330.46015)] or [\textit{H. Bendová}, J. Math. Anal. Appl. 412, No. 2, 1097--1104 (2014; Zbl 1322.46008)].) The main result states that, up to a subsequence, \((f_n)\) is either \(\varepsilon\)-weakly Cauchy on \(K\) or gives rise to almost \((\varepsilon/\pi)\)-copies of \(\ell^1\); more precisely, if \((f_n)\) is as above and if \(\varepsilon>0\), then \((f_n)\) admits a subsequence which is \(\varepsilon\)-weakly Cauchy on \(K\) or, for every \(\eta>0\), there is a subsequence \((f_{n_m})\) such that, if \(I\subset\mathbb{N}\) is finite and \(\phi:I\to\mathbb{N}\setminus I\) is any map, then \[ \left\|\sum_{m\in I}\lambda_m(f_{n_m}-f_{n_{\phi(m)}})\right\|\geq(\frac{\varepsilon}{\pi}-\eta)\sum_{m\in I}|\lambda_m| \] for all complex scalars \(\lambda_m\). The constant \(1/\pi\) improves Behrends's constant \(\sqrt{2}/8\). (Practical hint: the proof uses the third edition of \textit{W. Rudin}'s book [Real and complex analysis. 3rd ed. New York, NY: McGraw-Hill (1987; Zbl 0925.00005)], not the 1966 edition.) It is open whether \(1/\pi\) is optimal while for real scalars \(\lambda_m\) the optimal constant is \(1/2\). In a similar vein, Lebesgue's dominated convergence theorem (or, more generally, Rainwater's theorem) is quantified: Let \(X=C(K)\) (or, more generally, let \(X\) be a Banach space and \(K\) be a \(w^*\)-compact norming subset of the dual unit ball of \(X\)). Then any bounded sequence which is \(\varepsilon\)-weakly Cauchy on \(K\) has subsequences which are almost \(2\varepsilon\)-weakly Cauchy on the dual unit ball of \(X\).
scientific article; zbMATH DE number 6509536
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