On the best coapproximation problem in ℓ1n,

Abstract

We study the best coapproximation problem in the Banach space ${\ell }_1^n,$ by using Birkhoff–James orthogonality techniques. Given a subspace $\mathbb{Y}$ of ${\ell }_1^n$, we completely identify the elements x in ${\ell }_1^n,$ for which best coapproximations to x out of $\mathbb{Y}$ exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of ${\ell }_1^n$.

Keywords:

• Best coapproximations
• coproximinal subspace
• co-Chebyshev subspace
• Birkhoff–James orthogonality
• dual space

1991 Mathematics Subject Classifications:

• Primary 46B20
• Secondary 47L05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of Dr Debmalya Sain is supported by grant PID2021-122126NB-C31 funded by MCIN/AEI and by ‘ERDF A way of making Europe’ under the mentorship of Professor Miguel Martín. The second and third author would like to thank CSIR, Govt. of India, for the financial support in the form of Junior Research Fellowship under the mentorship of Prof. Kallol Paul.