Abstract

Following R. C. James' approach, we shall define the Banach space J(e)J(e) for each vector e=(e1,e2,...,ed)∈Rde=(e_1,e_2,...,e_d) \in \Bbb{R}^d with e1≠0e_1 \ne 0. The construction immediately implies that J(1)J(1) coincides with the Hilbert space l2l_2 and that J(1;−1)J(1;-1) coincides with the celebrated quasireflexive James space JJ. The results of this paper show that, up to an isomorphism, there are only these two possibilities: (i) J(e)J(e) is isomorphic to l2l_2 if e1+e2+...+ed≠0e_1+e_2+...+e_d\ne 0, and (ii) J(e)J(e) is isomorphic to JJ if e1+e2+...+ed=0e_1+e_2+...+e_d =0. Such a dichotomy also holds for every separable Orlicz sequence space lMl_M.

Author information

Contact details are reproduced from the original publication and may be historical.

Dušan Repovš

Faculty of Mathematics and Physics, University of Ljubljana, P. O. Box 2964, Ljubljana 1001, Slovenia

dusan.repovs@guest.arnes.si

Pavel V. Semenov

Department of Mathematics, Moscow City Pedagogical University, 2-nd Selskokhozyastvennyi pr. 4, Moscow 129226, Russia

pavels@orc.ru

Keywords

  • Hilbert space
  • Banach space
  • James sequence space
  • quasireflexive space
  • invertible continuous operator
  • Orlicz function

Mathematics Subject Classification

54C60, 54C65, 41A65; 54C55, 54C20

References

  1. R. C. James: Bases and reflexivity of Banach spaces, Ann. Math. (2) 52 (1950) 518–527.https://doi.org/10.2307/1969430Google Scholar
  2. R. C. James: A non-reflexive Banach space isometric with its second conjugate space, Proc. Natl. Acad. Sci. USA 37 (1951) 174–177.https://doi.org/10.1073/pnas.37.3.174Google Scholar
  3. H. Fetter, B. G. de Buen: The James Forest, London Math. Soc. Lecture Note Series 236, Cambridge University Press, Cambridge (1997).https://doi.org/10.1017/CBO9780511662379Google Scholar
  4. J. Lindenstrauss, C. Stegall: Examples of separable spaces which do not contain l1 and whose duals are not separable, Stud. Math. 54 (1975) 81–105.https://doi.org/10.4064/sm-54-1-81-105Google Scholar
  5. J. Lindenstrauss, L. Tzafriri: Classical Banach Spaces Vols 1 and 2, Springer, Berlin (1996).https://doi.org/10.1007/978-3-662-53294-2Google Scholar
  6. R. H. Lohman, P. G. Casazza: A general construction of spaces of the type of R. C. James, Can. J. Math. 27 (1975) 1263–1270.https://doi.org/10.4153/CJM-1975-131-3Google Scholar
  7. V. Meshkov: On smooth functions in the James space, Vestn. Mosk. Univ., Ser. Math 4 (1974) 9–13 (in Russian).Google Scholar
  8. B. S. Mityagin, I. S. Edelshtein: Homotopy type of linear groups of two classes of Banach spaces, Funct. Anal. Appl. 4(3) (1970) 61–72.https://doi.org/10.1007/BF01075242Google Scholar
  9. P. V. Semenov: James-Orlicz spaces, Russ. Math. Surv. 34(4) (1979) 185–186.https://doi.org/10.1070/RM1979v034n04ABEH003258Google Scholar
  10. P. V. Semenov: A counterexample to a Geoghegan-West problem, Proc. Amer. Math. Soc. 124 (1996) 939–943.https://doi.org/10.1090/S0002-9939-96-03276-5Google Scholar

Suggested citation

D. Repovš, P. V. Semenov. “A Unified Construction Yielding Precisely Hilbert and James Sequences Spaces.” Journal of Convex Analysis 17 (2010), No. 1, 349–356. https://doi.org/10.68381/jca17025

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.