Abstract
We study the best coapproximation problem in Banach spaces, by using Birkhoff–James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets–Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
References
Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935)
Bruck, R.E., Jr.: Property of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)
Bruck, R.E., Jr.: Nonexpansive projections onto subsets of Banach spaces. Pacific. J. Math. 47(2), 341–355 (1973)
Chmieliński, J.: On an \(\epsilon \)-Birkhoff orthogonality. J. Inequalities Pure Appl. Math. 6(3), 79 (2005)
Chmieliński, J.: Approximate Birkhoff–James Orthogonality in Normed Linear Spaces and Related Topics, Operator and norm inequalities and related topics, 303–320. Trends in Math, Birkhäuser/Springer, Cham (2022)
Dragomir, S.S.: On approximation of continuous linear functionals in normed linear spaces. An. Univ. Timişoara Ser. Ştiinţ. Mat. 29, 51–58 (1991)
Franchetti, C., Furi, M.: Some characteristic properties of real Hilbert spaces. Rev. Roumaine Math. Pures Appl. 17, 1045–1048 (1972)
Giles, J.R.: Strong differentiability of the norm and rotundity of the dual. J. Austral. Math. Soc. Ser. A 26, 302–308 (1978)
Ho, C., Zimmerman, S.: On the number of regions in an \(m\)-dimensional space cut by \(n\) hyperplanes. Austral. Math. Soc. Gaz. 33, 260–264 (2006)
James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947)
James, R.C.: Inner products in normed linear spaces. Bull. Am. Math Soc. 53, 559–566 (1947)
Kamiska, A., Lewicki, G.: Contractive and optimal sets in Banach spaces. Math. Nachr. 268, 74–95 (2004)
Lewicki, G., Trombetta, G.: Optimal and one-complemented subspaces. Monatsh. Math. 153, 115–132 (2008)
Megginson, R.E.: An Introduction to Banach Space Theory, Graduate Texts in Mathematics, 183 Springer, New York (1998)
Narang, T.D.: On best coapproximation in normed linear spaces. Rocky Mountain J. Math. 22, 265–287 (1992)
Papini, P.L., Singer, I.: Best coapproximation in normed linear spaces. Monatsh. Math. 88, 27–44 (1979)
Rao, G.S., Swaminathan, M.: Best coapproximation and schauder bases in Banach spaces. Acta Sci. Math. 54, 339–359 (1990)
Rudin, W.: Functional Analysis. McGraw-Hill, Inc (1991)
Sain, D., Paul, K., Bhunia, P., Bag, S.: On the numerical index of polyhedral Banach space. Linear Algebra Appl. 577, 121–133 (2019)
Sain, D., Sohel, S., Ghosh, S., Paul, K.: On best coapproximations in subspaces of diagonal matrices. Linear Multilinear Algebra 71, 47–62 (2023)
Sain, D., Sohel, S., Ghosh, S., Paul, K.: On Best Coapproximation Problem in \(\ell _1^n\). Linear Multilinear Algebra 72, 31–49 (2024)
Westphal, U.: Cosuns in \(\ell _p(n)\). J. Approx. Theory 54, 287–305 (1988)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Financial or non-financial competing interests
There are no relevant financial or non-financial competing interests to report.
Additional information
Communicated by Gerald Teschl.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
S. Sohel and S. Ghosh would like to thank CSIR, Govt. of India, for the financial support in the form of Senior Research Fellowship under the mentorship of Prof. Kallol Paul.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sohel, S., Ghosh, S., Sain, D. et al. On some special subspaces of a Banach space, from the perspective of best coapproximation. Monatsh Math (2024). https://doi.org/10.1007/s00605-023-01930-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00605-023-01930-2