Strongly sequentially continuous functions

A. Borichev, R. Deville, E. Matheron

Research output: Contribution to journalArticlepeer-review


Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).

Original languageEnglish (US)
Pages (from-to)535-548
Number of pages14
JournalQuaestiones Mathematicae
Issue number4
StatePublished - Dec 2001
Externally publishedYes


  • Completeness
  • Dirichlet’s theorem
  • Schur property
  • Topological groups

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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