Abstract
LetG be a separable complete metric additive topological group; it is shown that if a series Σx n is subseries convergent in any weaker Hausdorff group topology onG, then Σx n converges inG. This result can be used to obtain various extensions of the classical Orlicz-Pettis Theorem on subseries convergence in locally convex spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. F. Bachelis and H. P. Rosenthal,On unconditionally converging series and biorthogonal systems in a Banach space, Pacific J. Math.,37 (1971), 1–6.
S. Banach,Théorie des Operations Linéares, Warsaw, 1932.
G. Bennett and N. J. Kalton,FK-spaces containing c 0, to appear.
C. Bessaga and A. Pelczynski,On bases and unconditional convergence of series in Banach spaces, Studia Math.17 (1958), 151–164.
A. Grothendieck,Sur les applications linéares faiblement compactes d’espaces C(K), Canad. J. Math.5 (1953), 129–173.
N. J. Kalton,Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc.,70 (1971), 401–408.
J. L. Kelley,General Topology, New York, 1955.
G. Köthe,Topological Vector Spaces, Berlin, 1969.
C. W. McArthur,A note on subseries convergence, Proc. Amer. Math. Soc.12 (1961), 540–545.
C. W. McArthur,On a theorem of Orlicz and Pettis, Pacific J. Math.22 (1967), 297–302.
W. Orlicz,Beitrage zur Theorie der Orthogonalentwicklungen II, Studia Math.1 (1929), 241–255.
B. J. Pettis,On integration in vector spaces, Trans. Amer. Math. Soc.44 (1938), 277–304.
A. P. Robertson,On unconditional convergence in topological vector spaces, Proc. Roy. Soc. Edinburgh, Sect. A68 (1969), 145–157.
W. J. Stiles,On subseries convergence in F-spaces, Israel J. Math.8 (1970), 53–56.
E. Thomas,Sur le théoreme d’Orlicz et un problème de M. Laurent Schwartz, C. R. Acad. Sci. Paris. Sér. A.267 (1968), A7-A10.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kalton, N.J. Subseries convergence in topological groups and vector spaces. Israel J. Math. 10, 402–412 (1971). https://doi.org/10.1007/BF02771728
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02771728