A characterization of Banach spaces with separable duals via weak statistical convergence
Let \(A\subset \mathbb{N}\) and define the natural density of \(A\), \(\delta(A)\), as follows \[ \delta(A)= \lim_n n^{-1}|\{k\in A:k\leq n\}|, \] where \(|S|\) denotes the cardinality of the set \(S\). For a real or complex sequence \(x= (x_k)\) the term ``\(x\) converges to \(l\)'' means that for every \(\varepsilon> 0\) the set \[ \tau_x(\varepsilon):= \{k\in\mathbb{N}:|x_k-l|\geq \varepsilon\} \] is finite. If, instead of \(\tau_x(\varepsilon)\) being finite, we ask that \(\delta(\tau_x(\varepsilon))= 0\) for every \(\varepsilon> 0\) then we have the definition of what is called statistical convergence of \(x\) to \(l\). The notion of statistical convergence was first introduced by \textit{H. Fast} in his paper ``Sur la convergence statistique'' [Colloq. Math. 2, 241-244 (1951; Zbl 0044.33605)]. Statistical convergence in locally convex spaces was first introduced and investigated by \textit{I. J. Maddox}, in his paper ``Statistical convergence in a locally convex space'' [Math. Proc. Cambr. Philos. Soc. 104, No. 1, 141-145 (1988; Zbl 0674.40008)]. Let \((E,\tau)\) be a locally convex space whose topology \(\tau\) is generated by a family of seminorms \(Q\) and \(x= (x_k)\) be an \(E\)-valued sequence. Then \(x= (x_k)\) is said to be \(\tau\)-statistically convergent to \(l\in E\) if \(q(x_k-l)\) is statistically convergent to 0 for every \(q\in Q\). Following Maddox the authors define norm and weak statistical convergence. Definition. Let \(B\) be a Banach space, \((x_k)\) be a \(B\)-valued sequence, and \(x\in B\). 1. The sequence \((x_k)\) is norm statistically convergent to \(x\) provided that \(\delta(\{k:\|x_k- x\|> \varepsilon\})= 0\) for all \(\varepsilon> 0\). 2. The sequence \((x_k)\) is weakly statistically convergent to \(x\) provided that, for any \(x^*\) in the continuous dual \(B^*\) of \(B\), the sequence \((x^*(x_k- x))\) is statistically convergent to \(0\). The main result in the paper are the following: Theorem 1. Let \(B\) be a Banach space. Then \(B\) is finite-dimensional if and only if every weakly statistically null sequence has a bounded subsequence. Theorem 2. Let \(B\) be a separable Banach space. Then \(B\) has a separable dual if and only if every bounded weakly statistically null sequence agrees with a weakly null sequence on almost all indices. Theorem 3. Let \(B\) be a Banach space. Then \(B\) does not contain an isomorphic copy of \(l_1\) if and only if every bounded weakly statistically null sequence contains a weakly null subsequence.
scientific article; zbMATH DE number 1441401
scientific article