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)

Request Permissions   Purchase Content 
 
 

 

A characteristic property of the space $ s$


Author: Dietmar Vogt
Journal: Proc. Amer. Math. Soc. 143 (2015), 1183-1187
MSC (2010): Primary 46A45, 46A63, 46E10
DOI: https://doi.org/10.1090/S0002-9939-2014-12320-3
Published electronically: November 4, 2014
MathSciNet review: 3293733
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that under certain stability conditions a complemented subspace of the space $ s$ of rapidly decreasing sequences is isomorphic to $ s$ and this condition characterizes $ s$. This result is used to show that, for the classical Cantor set $ X$, the space $ C_\infty (X)$ of restrictions to $ X$ of $ C^\infty $-functions on $ \mathbb{R}$ is isomorphic to $ s$, which leads to an improvement of the theory developed in a previous work of the author.


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

  • [1] Bora Arslan, Alexander P. Goncharov, and Mefharet Kocatepe, Spaces of Whitney functions on Cantor-type sets, Canad. J. Math. 54 (2002), no. 2, 225-238. MR 1892995 (2003a:46041), https://doi.org/10.4153/CJM-2002-007-3
  • [2] A. Aytuna, J. Krone, and T. Terzioğlu, Complemented infinite type power series subspaces of nuclear Fréchet spaces, Math. Ann. 283 (1989), no. 2, 193-202. MR 980593 (90a:46014), https://doi.org/10.1007/BF01446430
  • [3] Leokadia Białas and Alexander Volberg, Markov's property of the Cantor ternary set, Studia Math. 104 (1993), no. 3, 259-268. MR 1220665 (94h:41024)
  • [4] A. Goncharov, Bases in the spaces of $ C^\infty $-functions on Cantor-type sets, Constr. Approx. 23 (2006), no. 3, 351-360. MR 2201471 (2006k:46036), https://doi.org/10.1007/s00365-005-0598-5
  • [5] Reinhold Meise and Dietmar Vogt, Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2, The Clarendon Press Oxford University Press, New York, 1997. Translated from the German by M. S. Ramanujan and revised by the authors. MR 1483073 (98g:46001)
  • [6] T. Terzioğlu, On the diametral dimension of some classes of $ F$-spaces, J. Karadeniz Univ. Fac. Arts Sci. Ser. Math.-Phys. 8 (1985), 1-13 (English, with Turkish summary). MR 924467 (88j:46006)
  • [7] Michael Tidten, Kriterien für die Existenz von Ausdehnungsoperatoren zu $ {\mathfrak{E}}(K)$ für kompakte Teilmengen $ K$ von $ {\bf R}$, Arch. Math. (Basel) 40 (1983), no. 1, 73-81 (German). MR 720896 (85g:46043), https://doi.org/10.1007/BF01192754
  • [8] D. Vogt, Structure theory of power series spaces of infinite type, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 339-363 (English, with English and Spanish summaries). MR 2068185 (2005e:46011)
  • [9] Dietmar Vogt, Restriction spaces of $ A^\infty $, Rev. Mat. Iberoam. 30 (2014), no. 1, 65-78. MR 3186931, https://doi.org/10.4171/RMI/769
  • [10] Dietmar Vogt and Max Josef Wagner, Charakterisierung der Quotientenräume von $ s$ und eine Vermutung von Martineau, Studia Math. 67 (1980), no. 3, 225-240 (German, with English summary). MR 592388 (81k:46002)
  • [11] Ahmed Zériahi, Inegalités de Markov et développement en série de polynômes orthogonaux des fonctions $ C^\infty $ et $ A^\infty $, Several complex variables (Stockholm, 1987/1988) Math. Notes, vol. 38, Princeton Univ. Press, Princeton, NJ, 1993, pp. 683-701 (French). MR 1207888 (94a:41018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46A45, 46A63, 46E10

Retrieve articles in all journals with MSC (2010): 46A45, 46A63, 46E10


Additional Information

Dietmar Vogt
Affiliation: FB Math.-Nat., Bergische Universität Wuppertal, Gauß-Str. 20, 42119 Wuppertal, Germany
Email: dvogt@math.uni-wuppertal.de

DOI: https://doi.org/10.1090/S0002-9939-2014-12320-3
Keywords: Space $s$, stability condition, Cantor set
Received by editor(s): May 16, 2013
Received by editor(s) in revised form: July 10, 2013
Published electronically: November 4, 2014
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society