Angelic spaces: Eberlein-Šmulian type results for abelian topological groups.

The flamboyant name of “angelic spaces” is attributed to a class of spaces in which different definitions of compactness are equivalent. The essential model of angelic spaces is given by metric spaces, since compactness, countable compactness and sequential compactness coincide for any subset of a metric space. The Eberlein-Šmulian Theorem establishes that such an equivalence also holds for the class of Banach spaces endowed with their weak topology.

During the last 50 years, many important, well-known theorems of Functional Analysis have been generalized from their topological vector space setting to the wider class of abelian topological groups. In this lecture we shall follow the trace of the Eberlein-Šmulian Theorem in the framework of topological abelian groups. We shall also report on more sophisticated sorts of compactness which might follow this pattern.

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[2] M. J. Chasco, E. Martín-Peinador and V. Tarieladze, A class of angelic sequential non-Fréchet-Urysohn topological groups, Topology and its Applications. 154 (2007) 741-748

[3] J. Orihuela, Pointwise compactness in spaces of continuous functions, J. London Math. Soc. (2) 36 (1987), no. 1, 143152.