Topological type of weakly closed subgroups in Banach spaces

Tadeusz Dobrowolski; Janusz Grabowski; Kazuhiro Kawamura

Studia Mathematica (1996)

  • Volume: 118, Issue: 1, page 49-62
  • ISSN: 0039-3223

Abstract

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The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in ℓ 1 which are interesting from the Banach space theory point of view are discussed.

How to cite

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Dobrowolski, Tadeusz, Grabowski, Janusz, and Kawamura, Kazuhiro. "Topological type of weakly closed subgroups in Banach spaces." Studia Mathematica 118.1 (1996): 49-62. <http://eudml.org/doc/216263>.

@article{Dobrowolski1996,
abstract = {The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in $ℓ^1$ which are interesting from the Banach space theory point of view are discussed.},
author = {Dobrowolski, Tadeusz, Grabowski, Janusz, Kawamura, Kazuhiro},
journal = {Studia Mathematica},
keywords = {additive subgroup of linear space; weakly closed; topological dimension; complete Erdős space; Lelek fan; additive subgroups; separable reflexive Banach spaces; complete Erdös space},
language = {eng},
number = {1},
pages = {49-62},
title = {Topological type of weakly closed subgroups in Banach spaces},
url = {http://eudml.org/doc/216263},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Dobrowolski, Tadeusz
AU - Grabowski, Janusz
AU - Kawamura, Kazuhiro
TI - Topological type of weakly closed subgroups in Banach spaces
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 1
SP - 49
EP - 62
AB - The main result says that nondiscrete, weakly closed, containing no nontrivial linear subspaces, additive subgroups in separable reflexive Banach spaces are homeomorphic to the complete Erdős space. Two examples of such subgroups in $ℓ^1$ which are interesting from the Banach space theory point of view are discussed.
LA - eng
KW - additive subgroup of linear space; weakly closed; topological dimension; complete Erdős space; Lelek fan; additive subgroups; separable reflexive Banach spaces; complete Erdös space
UR - http://eudml.org/doc/216263
ER -

References

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  1. [ADG] F. D. Ancel, T. Dobrowolski and J. Grabowski, Closed subgroups in Banach spaces, Studia Math. 109 (1994), 277-290. Zbl0840.46012
  2. [BP] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975. 
  3. [BO] W. T. Bula and L. G. Oversteegen, A characterization of smooth Cantor bouquets, Proc. Amer. Math. Soc. 108 (1990), 529-534. Zbl0679.54034
  4. [Ch] W. J. Charatonik, The Lelek fan is unique, Houston J. Math. 15 (1989), 27-34. Zbl0675.54034
  5. [D] T. Dobrowolski, Examples of topological groups homeomorphic to ℓ f 2 , Proc. Amer. Math. Soc. 98 (1986), 303-311. Zbl0605.22001
  6. [Da] M. M. Day Normed Linear Spaces, 3rd ed., Springer, Berlin, 1973. 
  7. [DG] T. Dobrowolski and J. Grabowski, Subgroups of Hilbert spaces, Math. Z. 211 (1992), 657-669. 
  8. [E] R. Engelking, Dimension Theory, PWN, Warszawa, and North-Holland, Amsterdam, 1978. 
  9. [KOT] K. Kawamura, L. G. Oversteegen and E. D. Tymchatyn, On homogeneous totally disconnected 1-dimensional spaces, Fund. Math., to appear. Zbl0861.54028
  10. [Le] A. Lelek, On plane dendroids and their endpoints in the classical sense, Fund. Math. 49 (1961), 301-319. Zbl0099.17701
  11. [Sie] W. Sierpiński, Sur la valeur asymptotique d'une certaine somme, Bull. Acad. Sci. Cracovie Math. Ser. A (1910), 11. Zbl41.0282.01

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