BEST COAPPROXIMATION IN METRIC LINEAR SPACES
Main Article Content
Abstract
In order to obtain some characterizations of real Hilbert spaces among real Banach spaces, a new kind of approximation, called best coapproximation, was introduced in normed linear spaces by C. Franchetti and M. Furi [3] in 1972. Subsequently, the study was pursued in normed linear spaces and Hilbert spaces by H. Berens, L. Hetzelt, T. D. Narang, P. L. Papini, Gectha S. Rao and her students, Ivan Singer and a few others (see, e.g., [1], [4], [7], [9], [13 to 15], and [17 to 20]). In this paper, we discuss best coapproximation in metric linear spaces thereby generalizing some of the results proved in [3], [7], [13], and [18]. The problems considered are those of existence of elements of best coapproximation and their characterization, characteriza tions of coproximinal, co-semi-Chebyshev and co-Chebyshev subspaces, and some properties of the best coapproximation map in metric linears space.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
H. Berens and U. Westphal, On the best coapproximation in a Hilbert space, in ' Quantitative Approximation' ( R. A. DeVore and K. Scherer, Eds., Academic Press, New York (1980), 7-10.
G. Birkhoff, Orthogonality in linear metric spaces, Duke Math J. 1(1935), 169-172.
C. Franchetti and M. Furi, Some characteristic properties of real Hilbert spaces, Rev. Roum Math. Pures Appl. 17(1972), 1045-1048.
L. Hetzelt, On suns and cosuns in finite dimensional normed real vector spaces, Acta Math Hung. 45(1985), 53-68.
J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions, Trans. Amer. Math. Soc. 148(1970), 147-169.
T. D. Narang, On certain characterizations of best approximation in metric linear spaces, Pure Appl. Mathematika Sciences, 4(1976), 121-124.
T. D. Narang, On best coapproximation, Ranchi Univ. Math. J. 17(1986), 49-56.
T. D. Narang, Best approximation in metric linear spaces, Math. Today, 5(1987), 21-28
T. D. Narang, On best coapproximation in normed linear spaces, Rocky Mountain J. Math., 22(1991), 265-287.
T. D. Narang, Best coapproximation in metric spaces, Puhl. Inst. Math. 51(1992), 71-76.
T. D. Narang and S. P. Singh, Best coapproximation in locally convex spaces, Tamkang J. Math 28(1997), 1-5.
G. Pantelidis, Approximationstheorie fur metrische lineare Raume, Math. Annal. 184 (1969), 30-48.
P. L. Papini and I. Singer, Best approximation in normed linear spaces, Math. Mh 88(1979), 27-44.
Geetha S. Rao, Best coapproximation in normed linear spaces: Approximation Theory, Vol V, Eds. Chui, Schumaker and Ward, Academic Press, New York (1986), 535-538.
Geetha S. Rao and K. R.·Chandrasekran, Characterisations of elements of best coapproxi mation in normed linear spaces, Pure Appl. Mathematika Scincccs 26(1987), 139-147.
Gcetha S. Rao and S. Elumalai, Approximation and strong approximation in locally convex spaces, Pure Appl. Mathematika Sciences 19(1984), 13-26.
Geetha S. Rao and S. Elumalai, Semi-continuity properties of the strong best coapproximation operator, Indian J. Pure Appl. Math. 16(1985), 257-270.
Gcctha S. Rao and S. Muthukumar, Semi-continuity properties of the coapproximation operator, Math. Today, 5(1987), 37-48
I. Singer, Best approximation in normed linear spaces by elements of linear subspaces, Springer-Verlag, New York, 1970.
U. Westphal, Cosuns in $ell^p(n)$, $1 le p < infty $, J. Approx. Theory, 54(1988), 287-305.