The uniform convergence ordinal index and the \(l^1\)-behavior of a sequence of functions (Q1826895)

\textit{W.~Szlenk} [Stud. Math. 30, 53--61 (1968; Zbl 0169.15303)] and \textit{J.~Bourgain} [Proc. Am. Math. Soc. 79, 241--246 (1980; Zbl 0438.46005)] showed the power of certain ordinal indices when applied to problems in the geometry of Banach spaces. A number of authors have subsequently used these and created new indices to great advantage. In the present paper, two such indices are defined. One is the index of uniform convergence \(\xi^{(f_n)}\) of a sequence of real valued functions \((f_n)\) on a set \(\Gamma\) which is pointwise convergent to a function \(f\). For \(\varepsilon >0\), set \[ U_\varepsilon^{(f_n)} = \{ F\in [\mathbb N]^{<\omega} : \exists\;\gamma\in\Gamma \;\text{ with }\;| f_i(\gamma) - f(\gamma)| \geq\varepsilon \;\forall\,i\in F\}. \] If this set is not pointwise closed for some \(\varepsilon>0\), then \(\xi^{(f_n)} \equiv \omega_1\). Otherwise \[ \xi^{(f_n)} \equiv \sup \{s_M(U_\varepsilon^{(f_n)}) :M\in [\mathbb N]^\omega,\;\varepsilon>0\}, \] where \(s_M\) denotes the strong Cantor-Bendixson index. The author proves that if \((f_n)\) is a sequence of uniformly bounded functions converging pointwise to \(f\), then \(\xi^{(f_n)} <\omega \) if and only if \((f_n-f)\) does not have an \(\ell_+^1\)-subsequence (in \(\ell^\infty(\Gamma)\)) if and only if \((f_n)\) converges weakly to \(f\) (in \(\ell^\infty(\Gamma)\)). One corollary is that if \((x_n)\) is a weak Cauchy sequence in a Banach space, then \((x_n)\) is weakly convergent if and only if \(\xi^{(x_n)} <\omega_1\), where the index is relative to \(\Gamma =\) unit ball of \(X^*\). Other quantified results are obtained concerning higher order \(\ell_1\) spreading models and the thin Schreier sets.