Structure of Sets of Strong Subdifferentiability in Dual L^1-Spaces

C. R. Jayanarayanan; T. S. S. R. K. Rao

doi:10.68381/jca28063

Abstract

We analyse the structure of finite dimensional subspaces of the set of points of strong subdifferentiability in a dual space. In a dual

L_1(\mu)

space, such a subspace is in the discrete part of the Yoshida-Hewitt type decomposition. In this set up, any Banach space consisting of points of strong subdifferentiability is necessarily finite dimensional. Our results also lead to streamlined and new proofs of results from the study of strong proximinality for subspaces of finite co-dimension in a Banach space.

Author information

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

C. R. Jayanarayanan

Department of Mathematics, Indian Institute of Technology, Palakkad, India

crjayan@iitpkd.ac.in

T. S. S. R. K. Rao

Department of Mathematics, Ashoka University, Sonipat, India

srin@fulbrightmail.org

Keywords

Strong subdifferentiability
strong proximinality
M-ideals
L-1-predual space

Mathematics Subject Classification

41A65; 46B20, 41A50, 46E15

References

M. Baronti: Norm-one projections onto subspaces of l∞, Arch. Math. (Basel) 51/3 (1988) 242–246.https://doi.org/10.1007/BF01207478 Google Scholar
S. Dutta, D. Narayana: Strongly proximinal subspaces of finite codimension in C(K), Colloq. Math. 109/1 (2007) 119–128.https://doi.org/10.4064/cm109-1-10 Google Scholar
M. Feder: On the sum of proximinal subspaces, J. Approx. Theory 49/2 (1987) 144–148.https://doi.org/10.1016/0021-9045(87)90084-0 Google Scholar
C. Franchetti, R. Payá: Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B(7) 7/1 (1993) 45–70.Google Scholar
G. Godefroy, V. Indumathi: Strong proximinality and polyhedral spaces, Rev. Mat. Complutense 14/1 (2001) 105–125.https://doi.org/10.5209/rev_REMA.2001.v14.n1.17047 Google Scholar
P. Harmand, D. Werner, W. Werner: M -Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer, Berlin (1993).https://doi.org/10.1007/BFb0084355 Google Scholar
V. Indumathi, S. Lalithambigai: A new proof of proximinality for M -ideals, Proc. Amer. Math. Soc. 135/4 (2007) 1159–1162.https://doi.org/10.1090/S0002-9939-06-08701-6 Google Scholar
C. R. Jayanarayanan: Proximinality properties in Lp (µ, X) and polyhedral direct sums of Banach spaces, Numer. Funct. Anal. Optim. 35/6 (2014) 708–723.https://doi.org/10.1080/01630563.2013.841195 Google Scholar
C. R. Jayanarayanan, T. Paul: Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl. 426/2 (2015) 1217–1231.https://doi.org/10.1016/j.jmaa.2015.01.013 Google Scholar
H. E. Lacey: The Isometric Theory of Classical Banach Spaces, Grundlehren der mathematischen Wissenschaften 208, Springer, New York (1974).https://doi.org/10.1007/978-3-642-65762-7 Google Scholar
P.-K. Lin: A remark on the sum of proximinal subspaces, J. Approx. Theory 58/1 (1989) 55–57.https://doi.org/10.1016/0021-9045(89)90007-5 Google Scholar
T. S. S. R. K. Rao: Points of strong subdifferentiability in dual spaces, Houston J. Math. 44/4 (2018) 1221–1226.Google Scholar
W. Rudin: Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York (1991).Google Scholar
L. P. Vlasov: Approximate properties of subspaces of finite codimension in C(Q), Mat. Zametki 28/2 (1980) 205–222.https://doi.org/10.1007/BF01157916 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2021. Rights now held by Banach Press.

[1] M. Baronti: Norm-one projections onto subspaces of l∞, Arch. Math. (Basel) 51/3 (1988) 242–246.https://doi.org/10.1007/BF01207478 Google Scholar

[2] S. Dutta, D. Narayana: Strongly proximinal subspaces of finite codimension in C(K), Colloq. Math. 109/1 (2007) 119–128.https://doi.org/10.4064/cm109-1-10 Google Scholar

[3] M. Feder: On the sum of proximinal subspaces, J. Approx. Theory 49/2 (1987) 144–148.https://doi.org/10.1016/0021-9045(87)90084-0 Google Scholar

[4] C. Franchetti, R. Payá: Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B(7) 7/1 (1993) 45–70.Google Scholar

[5] G. Godefroy, V. Indumathi: Strong proximinality and polyhedral spaces, Rev. Mat. Complutense 14/1 (2001) 105–125.https://doi.org/10.5209/rev_REMA.2001.v14.n1.17047 Google Scholar

[6] P. Harmand, D. Werner, W. Werner: M -Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer, Berlin (1993).https://doi.org/10.1007/BFb0084355 Google Scholar

[7] V. Indumathi, S. Lalithambigai: A new proof of proximinality for M -ideals, Proc. Amer. Math. Soc. 135/4 (2007) 1159–1162.https://doi.org/10.1090/S0002-9939-06-08701-6 Google Scholar

[8] C. R. Jayanarayanan: Proximinality properties in Lp (µ, X) and polyhedral direct sums of Banach spaces, Numer. Funct. Anal. Optim. 35/6 (2014) 708–723.https://doi.org/10.1080/01630563.2013.841195 Google Scholar

[9] C. R. Jayanarayanan, T. Paul: Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl. 426/2 (2015) 1217–1231.https://doi.org/10.1016/j.jmaa.2015.01.013 Google Scholar

[10] H. E. Lacey: The Isometric Theory of Classical Banach Spaces, Grundlehren der mathematischen Wissenschaften 208, Springer, New York (1974).https://doi.org/10.1007/978-3-642-65762-7 Google Scholar

[11] P.-K. Lin: A remark on the sum of proximinal subspaces, J. Approx. Theory 58/1 (1989) 55–57.https://doi.org/10.1016/0021-9045(89)90007-5 Google Scholar

[12] T. S. S. R. K. Rao: Points of strong subdifferentiability in dual spaces, Houston J. Math. 44/4 (2018) 1221–1226.Google Scholar

[13] W. Rudin: Functional Analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, New York (1991).Google Scholar

[14] L. P. Vlasov: Approximate properties of subspaces of finite codimension in C(Q), Mat. Zametki 28/2 (1980) 205–222.https://doi.org/10.1007/BF01157916 Google Scholar