Abstract
We present a generalization of the classical Orlicz–Pettis theorem about subseries convergence in topological vector spaces. In preparation we review some aspects of the theory of locally convex cones, a generalization of locally convex topological vector spaces. We introduce conical extensions of the classical sequence spaces and a version of Schur’s theorem about weak and norm convergence of sequences in the extension of \(\ell ^1.\) Our version of the Orlicz–Pettis theorem refers to series of bounded elements of a locally convex cone.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
No datasets were generated or analysed during the current study.
References
Drewnowski, L., Labuda, I.: The Orlicz–Pettis theorem for for topological Riesz spaces Proc. Am. Math. Soc. 126(3), 823–825 (1998)
Filter, W., Labuda, I.: Essay on the Orlicz–Pettis Theorem, I (the two Theorems) Real Analysis Exchange 16 vol. 2, pp. 393–403 (1990)
Keimel, K., Roth, W.: Ordered Cones and Approximation. Lecture Notes in Mathematics, vol. 1517. Springer Verlag, Heidelberg-Berlin-New York (1992)
McArthur, C.W.: A note on subseries convergence. Proc. Am. Math. Soc. 12, 540–545 (1961)
McArthur, C.W.: On a theorem of Orlicz and Pettis. Pac. J. Math. 22, 297–302 (1967)
Muscat, J.: Functional Analysis. Springer Verlag, Heidelberg (2014)
Orlicz, W.: Beiträge zur Theorie der Orthogonalentwicklungen II. Stud. Math. 1, 241–255 (1929)
Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44(2), 277–304 (1938)
Robertson, A.P.: On unconditional convergence in topological vector spaces. Proc. Roy. Soc. Edinburgh A 68, 145–157 (1969)
Roth, W.: Operator-Valued Measures and Integrals for Cone-Valued Functions. Lecture Notes in Mathematics, vol. 1964. Springer Verlag, Heidelberg-Berlin-New York (2009)
Roth, W.: Hahn–Banach type theorems for locally convex cones. J. Aust. Math. Soc. (Ser. A) 68(1), 104–125 (2000)
Schur, I.: Gesammelte Abhandlungen. In: Brauer, H.A., Rohrbach, H. (eds.) Springer Collected Works in Mathematics. Springer Verlag, Heidelberg-Berlin-New York (1973)
Talagrand, M.: Pettis Integral and Measure Theory, Memoirs of the American Mathematical Society, 51 (1984)
Author information
Authors and Affiliations
Contributions
W.R. is single author and wrote the whole manuscrit.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Roth, W. The Orlicz–Pettis theorem for locally convex cones. Positivity 28, 75 (2024). https://doi.org/10.1007/s11117-024-01084-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11117-024-01084-x