Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 135-139

LIGHT SIDE OF COMPACTNESS IN LEBESGUE SPACES: SUDAKOV THEOREM

Przemyslaw Górka and Humberto Rafeiro

Warsaw University of Technology, Department of Mathematics and Information Sciences
Ul. Koszykowa 75, 00-662 Warsaw, Poland; p.gorka 'at' mini.pw.edu.pl

Pontificia Universidad Javeriana, Facultad de Ciencias, Departamento de Matemáticas
Cra. 7 No. 43-82, Bogotá, Colombia; silva-h 'at' javeriana.edu.co

Abstract. In this note we show that, in the case of bounded sets in metric spaces with some additional structure, the boundedness of a family of Lebesgue p-summable functions follow from a certain uniform limit norm condition. As a byproduct, the well known Riesz–Kolmogorov compactness theorem can be formulated only with one condition.

2010 Mathematics Subject Classification: Primary 46B50; Secondary 46E30.

Key words: Compactness, Riesz–Kolmogorov theorem, metric measure spaces.

Reference to this article: P. Górka and H. Rafeiro: Light side of compactness in Lebesgue spaces: Sudakov theorem. Ann. Acad. Sci. Fenn. Math. 42 (2017), 135-139.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4209