ArticleOriginal scientific text
Title
Closed subgroups in Banach spaces
Authors 1, 2, 3
Affiliations
- Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, U.S.A.
- Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
- Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Abstract
We show that zero-dimensional nondiscrete closed subgroups do exist in Banach spaces E. This happens exactly when E contains an isomorphic copy of . Other results on subgroups of linear spaces are obtained.
Keywords
additive subgroup of linear space, basic sequence, weakly closed, topological dimension
Bibliography
- [AO] J. M. Aarts and L. G. Oversteegen, The product structure of homogeneous spaces, Indag. Math. 1 (1990), 1-5.
- [BP1] C. Bessaga and A. Pełczyński, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151-164.
- [BP2] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa, 1975.
- [BPR] C. Bessaga, A. Pełczyński and S. Rolewicz, Some properties of the space (s), Colloq. Math. 7 (1957), 45-51.
- [BHM] R. Brown, P. J. Higgins and S. A. Morris Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties, Math. Proc. Cambridge Philos. Soc. 78 (1975), 19-32.
- [Da] M. M. Day, Normed Linear Spaces, 3rd ed., Springer, Berlin, 1973.
- [Di] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
- [DG] T. Dobrowolski and J. Grabowski, Subgroups of Hilbert spaces, Math. Z. 211 (1992), 657-669.
- [DT] T. Dobrowolski and H. Toruńczyk, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235.
- [E] R. Engelking, Dimension Theory, North-Holland, Amsterdam, 1978.
- [K] N. J. Kalton, Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151-177.
- [P] A. Pełczyński, Projections in certain Banach spaces, Studia Math. 19 (1960), 209-228.
- [T] H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), 247-262.