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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 $(x\sb k)\in X$ converges statistically to $\ell \in X$ if $n\sp{-1}\vert \{k\le n:\quad q(x\sb k-\ell)\ge \epsilon \}\vert \to 0$ as $n\to \infty$, $\forall q\in Q$, $\forall \epsilon >0$, where $\vert A\vert$ 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 $(x\sb k)$ for which $\exists \ell \in X$ such that $(1/n)\sum\sp{n}\sb{k=1}f(q(x\sb k-\ell))\to 0$ as $n\to \infty$, $\forall q\in Q$. At last one says that $(x\sb k)$ is slowly oscillating if $(x\sb k-x\sb n)\to 0$ as $k\to \infty$, $n\ge k$ and n/k$\to 1$. For every modulus f, the following results hold: 1) $[x\sb k\to \ell (w(f))]\ \Rightarrow\ [x\sb k\to \ell (s)];$ 2) $[S\equiv w(f)]\ \Leftrightarrow\ [f$ is bounded] 3) $[x\sb k\to \ell (w(f))] \bigwedge [(x\sb k)$ is slowly oscillating]$\ \Rightarrow\ [x\sb k\to \ell].$ If X is a Banach space, and $w\sb 1$ is the space $w(f)$ with $f(t)=1$, then 4) $[w(f)$ is locally convex]$\ \Leftrightarrow\ [w(f)=w\sb 1]\ \Leftrightarrow\ [\lim\sb{t\to \infty}f(t)/t>0].$
Reviewer: F.Barbieri

40J05Summability in abstract structures
46A45Sequence spaces
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