<abstract xmlns="http://www.w3.org/1999/xhtml"> We investigate the subsets of the Fr´echet space <italic>s </italic>of all sequences of real numbers equipped with the Fr´echet metric <italic>ρ </italic>from the Baire category point of view. In particular, we concentrate on the “convergence” sets of the series ∑ƒ&lt;sub><italic>n&lt;/sub> </italic>(<italic>x&lt;sub>n&lt;/sub></italic>) that is, sets of sequences <italic>x </italic>= (<italic>x&lt;sub>n&lt;/sub></italic>) for which the series converges, or has a sum (perhaps infinite), or oscillates. Provided all ƒ&lt;sub><italic>n&lt;/sub> </italic>are continuous real functions, sufficient conditions are given for the “convergence” sets to be of the first Baire category or residual in <italic>s</italic>.</abstract>

We investigate the subsets of the Fr´echet space s of all sequences of real numbers equipped with the Fr´echet metric ρ from the Baire category point of view. In particular, we concentrate on the “convergence” sets of the series ∑ƒn (xn) that is, sets of sequences x = (xn) for which the series converges, or has a sum (perhaps infinite), or oscillates. Provided all ƒn are continuous real functions, sufficient conditions are given for the “convergence” sets to be of the first Baire category or residual in s.

The structure of the fréchet space s regarding the series ∑ƒn(xn)

Mathematical Institute Slovak Academy of Sciences Štefánikova 49 SK–814-73 Bratislava SLOVAKIA

Tatra Mountains Mathematical Publications

The structure of the fréchet space s regarding the series ∑ƒ(x)

We investigate the subsets of the Fr´echet space s of all sequences of real numbers equipped with the Fr´echet metric ρ from the Baire category point of view. In particular, we concentrate on the “convergence” sets of the series ∑ƒn (xn) that is, sets of sequences x = (xn) for which the series converges, or has a sum (perhaps infinite), or oscillates. Provided all ƒn are continuous real functions, sufficient conditions are given for the “convergence” sets to be of the first Baire category or residual in s.

We investigate the subsets of the Fr´echet space s of all sequences of real numbers equipped with the Fr´echet metric ρ from the Baire category point of view....

The structure of the fréchet space s regarding the series ∑ƒ_n(x_n)

Publié en ligne: 12 nov. 2012

Pages: 1 - 8

DOI: https://doi.org/10.2478/v10127-009-0042-5

Mots clés
Fr´echet space, Baire category, continuous functions, residual set, series.

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