EQUIVALENCE RELATIONS ON BEST COAPPROXIMATION AND WORST COAPPROXIMATION

Document Type : Original Article

Authors
H. Mazaheri 1
S. M. Jesmani 2
1 Department of Mathematics, Yazd University, 89195-741, Yazd, Iran.
2 Department of Materials and Metallurgical Engineering, Technical and Vocational University(TVU), Tehran, Iran
10.22034/maco.3.2.11
Abstract
 A kind of approximation, called best coapproximation was introduced and discussed in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by several researchers in different abstract spaces. In this paper, we define relations on best coapproximation and worst coapproximation. We show that these relations are equivalence relation. We obtain cosets sets of best coapproximation and worst approximation. We obtain some results on these sets, compactness and weakly compactness and define coqproximinal and coqremotal.

Keywords

[1] G. BIRKttOFF, Orthogonality in linear metric spaces, Duke Math. J. 1, 169–172, 1935.
[2] JR. R. E. BRUCK, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47, 341–355 ,1973.
[3] E. W. CHENEY and K. H. PRICE Minimal projections. In : Approximation Theory (Ed. A. TALBOT), p. 261–289. New York: Academic Press. 1970.
[4] M. M. DAY, Review of the paper ofJ. R. HOLUB ”Rotundity, orthogonality and characterizations of inner product spaces” Math. Rev. 52, 175, 1976.
[5] C. FRANCHETTI and M. FURI Some characteristic properties of real Hilbert spaces Rev. Roumaine Math. Pures Appl. 17, 1045–1048 , 1972.
[6] J. R. HOLUB, Rotundity, orthogonality and characterizations of inner product spaces, Bull. Amer. Math. Soc. 81, 1087–1089, 1975.
[7] R. C. JAI fES, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc.61,265–292, 1947.
[8] A. LAZAR and M. ZIPPIN On finite-dimensional subspaces of Banach spaces, Israel J. Math. 3, 147–156, 1965.
[9] J. LINDENSTRAUSS, On projections with norm 1 – an example, Proe. Amer. Math. Soc. 15, 403-406, 1964.
[10] P. L. PAPINI, Some questions related to the concept of orthogonality in Banaeh spaces, Proximity maps; bases. Boll. Un. Mat. Itah (4) ll, 44–63 1975.
[11] P. L. PAPINI, Approximation and strong approximation in normed spaces via tangent functionals, J. Approx. Theory 22, 111–118 1978.
[12] I. SINGER, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces Bucharest: Publ. House Acad. and Berlin–Heidelberg–New York: Springer. 1970.
[13] I. SINGER, The theory of best approximation and functional analysis CBMS Reg. Conf. Ser. Appl. Math. 13. Philadelphia: SIAM, 1974.
[14] I. SINGER, Some classes of non-linear operators generalizing the metric projections onto ebyw subspaees, In: Proc. Fifth Internat. Summer School ”Theory of Nonlinear Operators”, held in Berlin 1977. Abhandh Akad. Wiss. DDR, Abt. Math. - Naturwiss. Technik 6 245–257, 1978.
[15] P. L. PAPANI, I. SINGER, . Best coapproximation in normed linear spaces, Monatsh. Math. 88 27–44, 1979.
Mathematical Analysis and Convex Optimization
Volume 3, Issue 2
December 2022
Pages 119-127