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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 46A45 Sequence spaces (including Köthe sequence spaces)
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##### References:
  DOI: 10.1007/BF01175646 · Zbl 0045.33403  DOI: 10.2307/2308747 · Zbl 0089.04002  Sal?t, Math. Slovaca 2 pp 139– (1980)  Fast, Colloq. Math 2 pp 241– (1951)  Maddox, Math. Proc. Cambridge Philos. Soc 100 pp 161– (1986)  Fridy, Analysis 5 pp 301– (1985) · Zbl 0588.40001  Maddox, Math. Proc. Cambridge Philos. Soc 101 pp 523– (1987)
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