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Statistical convergence in a locally convex space. (English) Zbl 0674.40008
Let X be a locally convex Hausdorff topological linear space, whose topology is given by a family Q of continuous seminorms q. A sequence \((x_ k)\in X\) converges statistically to \(\ell \in X\) if \(n^{-1}| \{k\leq n:\quad q(x_ k-\ell)\geq \epsilon \}| \to 0\) as \(n\to \infty\), \(\forall q\in Q\), \(\forall \epsilon >0\), where \(| A|\) denotes the cardinality of the set A. Let S be the space of sequences statistically convergent in X. Given a modulus f (for the definition see also the author [Math. Proc. Camb. Philos. Soc. 100, 161-166 (1986; Zbl 0631.46010)]), w(f) denotes the set of \((x_ k)\) for which \(\exists \ell \in X\) such that \((1/n)\sum^{n}_{k=1}f(q(x_ k-\ell))\to 0\) as \(n\to \infty\), \(\forall q\in Q\). At last one says that \((x_ k)\) is slowly oscillating if \((x_ k-x_ n)\to 0\) as \(k\to \infty\), \(n\geq k\) and n/k\(\to 1\). For every modulus f, the following results hold:
1) \([x_ k\to \ell (w(f))]\;\Rightarrow\;[x_ k\to \ell (s)];\)
2) \([S\equiv w(f)]\;\Leftrightarrow\;[f\) is bounded]
3) \([x_ k\to \ell (w(f))] \bigwedge [(x_ k)\) is slowly oscillating]\(\;\Rightarrow\;[x_ k\to \ell].\)
If X is a Banach space, and \(w_ 1\) is the space \(w(f)\) with \(f(t)=1\), then
4) \([w(f)\) is locally convex]\(\;\Leftrightarrow\;[w(f)=w_ 1]\;\Leftrightarrow\;[\lim_{t\to \infty}f(t)/t>0].\)
Reviewer: F.Barbieri

MSC:
40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX)
46A45 Sequence spaces (including Köthe sequence spaces)
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