## 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.

### MSC:

 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

### Citations:

Zbl 0444.33605; Zbl 0674.40008; Zbl 0044.33605
Full Text:

### References:

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