A characterization of Banach spaces with separable duals via weak statistical convergence. (English) Zbl 0982.46007

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 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 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.


46B10 Duality and reflexivity in normed linear and Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
40A05 Convergence and divergence of series and sequences
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