Principle of local reflexivity revisited
HTML articles powered by AMS MathViewer
- by Eve Oja and Märt Põldvere
- Proc. Amer. Math. Soc. 135 (2007), 1081-1088
- DOI: https://doi.org/10.1090/S0002-9939-06-08612-6
- Published electronically: October 2, 2006
- PDF | Request permission
Abstract:
We give, departing from Grothendieck’s description of the dual of the space of weak$^\ast$-weak continuous finite-rank operators, a clear proof for the principle of local reflexivity in a general form.References
- Ehrhard Behrends, A generalization of the principle of local reflexivity, Rev. Roumaine Math. Pures Appl. 31 (1986), no. 4, 293–296. MR 854826
- S. J. Bernau, A unified approach to the principle of local reflexivity, Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980, pp. 427–439. MR 606228
- Peter G. Casazza, Approximation properties, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 271–316. MR 1863695, DOI 10.1016/S1874-5849(01)80009-7
- Antonio Martínez-Abejón, An elementary proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1397–1398. MR 1476378, DOI 10.1090/S0002-9939-99-04687-0
- David W. Dean, The equation $L(E,\,X^{\ast \ast })=L(E,\,X)^{\ast \ast }$ and the principle of local reflexivity, Proc. Amer. Math. Soc. 40 (1973), 146–148. MR 324383, DOI 10.1090/S0002-9939-1973-0324383-X
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
- Hicham Fakhoury, Sélections linéaires associées au théorème de Hahn-Banach, J. Functional Analysis 11 (1972), 436–452 (French). MR 0348457, DOI 10.1016/0022-1236(72)90065-1
- G. Godefroy, N. J. Kalton, and P. D. Saphar, Unconditional ideals in Banach spaces, Studia Math. 104 (1993), no. 1, 13–59. MR 1208038
- Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
- P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, vol. 1547, Springer-Verlag, Berlin, 1993. MR 1238713, DOI 10.1007/BFb0084355
- William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 1–84. MR 1863689, DOI 10.1016/S1874-5849(01)80003-6
- W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. MR 280983, DOI 10.1007/BF02771464
- N. J. Kalton, Locally complemented subspaces and ${\cal L}_{p}$-spaces for $0<p<1$, Math. Nachr. 115 (1984), 71–97. MR 755269, DOI 10.1002/mana.19841150107
- Joram Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 112. MR 179580
- J. Lindenstrauss and H. P. Rosenthal, The ${\cal L}_{p}$ spaces, Israel J. Math. 7 (1969), 325–349. MR 270119, DOI 10.1007/BF02788865
- Antonio Martínez-Abejón, An elementary proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1397–1398. MR 1476378, DOI 10.1090/S0002-9939-99-04687-0
- Eve Oja, Geometry of Banach spaces having shrinking approximations of the identity, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2801–2823. MR 1675226, DOI 10.1090/S0002-9947-00-02521-6
- Eve Oja, Operators that are nuclear whenever they are nuclear for a larger range space, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 3, 679–694. MR 2097268, DOI 10.1017/S0013091502001165
- Raymond A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. MR 1888309, DOI 10.1007/978-1-4471-3903-4
- Charles Stegall, A proof of the principle of local reflexivity, Proc. Amer. Math. Soc. 78 (1980), no. 1, 154–156. MR 548105, DOI 10.1090/S0002-9939-1980-0548105-6
Bibliographic Information
- Eve Oja
- Affiliation: Institute of Pure Mathematics, Tartu University, J. Liivi 2, 50409 Tartu, Estonia
- Email: eve.oja@ut.ee
- Märt Põldvere
- Affiliation: Institute of Pure Mathematics, Tartu University, J. Liivi 2, 50409 Tartu, Estonia
- Email: mart.poldvere@ut.ee
- Received by editor(s): April 6, 2005
- Received by editor(s) in revised form: November 2, 2005
- Published electronically: October 2, 2006
- Additional Notes: This research was partially supported by Estonian Science Foundation Grant 5704
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1081-1088
- MSC (2000): Primary 46B07, 46B20, 46B28
- DOI: https://doi.org/10.1090/S0002-9939-06-08612-6
- MathSciNet review: 2262909