Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Affine and homeomorphic embeddings into $\ell ^{2}$


Authors: Czeslaw Bessaga and Tadeusz Dobrowolski
Journal: Proc. Amer. Math. Soc. 125 (1997), 259-268
MSC (1991): Primary 57N17
DOI: https://doi.org/10.1090/S0002-9939-97-03832-X
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that

(1)
a locally compact convex subset $C$ of a topological vector space that admits a sequence of continuous affine functionals separating points of $C$ affinely embeds into a Hilbert space;
(2)
an infinite-dimensional locally compact convex subset of a metric linear space has a central point;
(3)
every $\sigma $-compact locally convex metric linear space topologically embeds onto a pre-Hilbert space.


References [Enhancements On Off] (What's this?)

  • [B] C. Bessaga, Infinite-dimensional locally compact convex sets and the shapes of compacta, Bull. Acad. Polon. Sci., sér. sci. math. astr. et phys. 24 (1976), 589-591. MR 54:8640
  • [BP1] C. Bessaga and A. Pelczynski, On spaces of measurable functions, Studia Math. 44 (1972), 597-615. MR 51:4310
  • [BP2] -, Selected Topics in Infinite-Dimensional Topology, vol. 58, PWN MM, Warszawa, 1975. MR 57:17657
  • [vBDHvM] J. van der Bijl, T. Dobrowolski, K.P. Hart and J. van Mill, Admissibility, homeomorphism extension and the $\operatorname {AR} $-property in topological linear spaces, Topology Appl. 48 (1992), 63-81. MR 94b:57024
  • [Ca] R. Cauty, Un space métrique linéare qui n'est pas un rétracte absolu, Fund. Math. 146 (1994), 85-99. MR 95j:54022
  • [CDM] D. Curtis, T. Dobrowolski and J. Mogilski, Some applications of the topological characterizations of the sigma-compact spaces $\ell ^{f}_{2}$ and $\Sigma $, Trans. A.M.S. 284 (1984), 847-846. MR 86i:54035
  • [Do1] T. Dobrowolski, An extension of a theorem of Klee, in Proc. Fifth Prague Topol. Sympos. (1981), 147-150. MR 84d:57008
  • [Do2] -, On extending mappings into nonlocally convex metric spaces, Proc. A.M.S. 93 (1985), 555-560. MR 86e:54023
  • [Do3] -, Extending homeomorphisms and applications to metric linear spaces without completeness, Trans. A.M.S. 313 (1989), 753-784. MR 89j:57010
  • [DM1] T. Dobrowolski and J. Mogilski, Problems on topological classification of incomplete metric spaces, Open Problems in Topology (J. van Mill and G.M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 409-429. CMP 91:03
  • [DM2] -, Regular retractions onto finite dimensional convex sets, Function Spaces: The Second Conference (K. Jarosz, ed.), Lectures Notes in Pure and Applied Mathematics 172, Marcel Dekker, New York, 1995, pp. 85-99. CMP 96:01
  • [DT1] T. Dobrowolski and H. Torunczyk, On metric linear spaces homeomorphic to $\ell ^2 $ and convex sets homeomorphic to $Q$, Bull. Acad. Polon. Sic., ser. math. astronom. phys. et math. 27 (1979), 883-887. MR 82j:57010
  • [DT2] -, Separable complete ANR's admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235. MR 83a:58007
  • [En] R. Engelking, General Topology, Polish Scientific Publishers, Warszawa, 1977. MR 58:18316b
  • [KPR] N.J. Kalton, N.T. Peck and J.W. Roberts, An $F$-space Sampler, London Math. Soc. Lecture Note Series 89, 1984. MR 87c:46002
  • [Kl] V.L. Klee, On the Borelian and projective types of linear subspaces, Math. Scand. 6 (1958), 189-199. MR 21:3752
  • [Mar] W. Marciszewski, On topological embeddings of linear metric spaces, Math. Ann. (to appear).
  • [NT] N.T. Nhu and L.H. Tri, Every needle point space contains a compact convex $\operatorname {AR} $-set with no extreme points, Proc. A.M.S. 120 (1994), 1261-1265. MR 94f:54038
  • [R1] J.W. Roberts, A compact convex set with no extreme points, Studia Math. 60 (1977), 255-266. MR 57:10595
  • [R2] -, Pathological compact convex sets in the spaces $L_{p}$, $0\leq p\leq 1$, The Altgeld Book, University of Illinois, 1976.
  • [Tor] H. Torunczyk, Concerning locally homotopy negligible sets and characterization of $\ell _{2}$-manifolds, Fund. Math. 101 (1978), 93-110. MR 80g:57019

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57N17

Retrieve articles in all journals with MSC (1991): 57N17


Additional Information

Czeslaw Bessaga
Affiliation: Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland
Email: bessaga@impan.impan.gov.pl

Tadeusz Dobrowolski
Affiliation: Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland
Address at time of publication: Department of Mathematics, Pittsburg State University, Pittsburg, Kansas 66762
Email: tdobrowo@mail.pittstate.edu

DOI: https://doi.org/10.1090/S0002-9939-97-03832-X
Keywords: Convex set, affine embedding, locally convex space, central points, $\sigma $-compact spaces
Received by editor(s): July 21, 1992
Communicated by: James E. West
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society