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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 )X converges statistically to X if n -1 |{kn:q(x k -)ϵ}|0 as n, qQ, ϵ>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 X such that (1/n) k=1 n f(q(x k -))0 as n, qQ. At last one says that (x k ) is slowly oscillating if (x k -x n )0 as k, nk and n/k1. For every modulus f, the following results hold:

1) [x k (w(f))][x k (s)];

2) [Sw(f)][f is bounded]

3) [x k (w(f))][(x k ) is slowly oscillating][x k ]·

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][w(f)=w 1 ][lim t f(t)/t>0]·

Reviewer: F.Barbieri

MSC:
40J05Summability in abstract structures
46A45Sequence spaces