Document Type : Research Paper

Authors

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.

Abstract

For a non-zero normed linear space $A$, we consider $ C^b\left(K\right) $, the complex-valued, bounded and continuous functions space on $K$ with $ \left\| \cdot \right\|_\infty $, where $ K = \overline{B^{\left(0\right)}_1} $ (the closed unit ball of $A$). Also for a non-zero element $\varphi \in A^*$ with $ \left\| \varphi \right\| \leq 1 $, we consider the space $ C^{b\varphi}\left(K\right) $ as the linear space $ C^b\left(K\right) $ with the new norm $ \left\| f \right\|_\varphi = \left\| f\varphi \right\|_\infty $ for all $ f \in C^b\left(K\right) $. Some basic properties such as, proximinality, E-proximinality, strongly proximinality and quasi Chebyshev for certain subsets of $ C^b\left(K\right) $ are characterized with the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\|\cdot \right\|_\infty $. Some examples for illustration and for comparison between the norms $ \left\| \cdot \right\|_\varphi $ and $ \left\| \cdot \right\|_\infty $ on $ C^b\left(K\right) $ are presented.

Keywords

[1] F. Deutsch, Best Approximation in Inner Product Spaces, Canad. Math. Soc., (2000).
[2] G. Godefroy and V. Indumathi, Strong proximinality and polyhedral spaces, Rev. Math. Complut., (2001), pp. 105-125.
[3] V. Indumathi, N. Prakash, E-proximinal subspaces, J. Math. Anal. Appl., 450 (2017), pp. 1-11.
[4] A.R. Khoddami, Weak and cyclic amenability of certain function algebras, Wavelets Linear Algebr., 7 (2020), pp. 31-41.
[5] A.R. Khoddami, Non-equivalent norms on $ C^b\left(K\right)$, Sahand Commun. Math. Anal., 4 (2020), pp. 1-11.
[6] D. Narayana and T.S.S.R.K. Rao, Some remarks on quasi-Chebyshev subspaces, J. Math. Anal. Appl., 321 (2006), pp. 193-197.
[7] T.S.S.R.K. Rao, Best approximation in spaces of compact operators, Linear Algebra Appl., 627 (2021), pp. 72-79.