Hamid Mazaheri Tehrani - Faculty of Mathematics, Yazd University, Yazd, Iran
N. Kh. Hamidi - Faculty of Mathematics, Yazd University, Yazd, Iran
M .J. Salehi - Faculty of Mathematics, Yazd University, Yazd, Iran

In this paper, we define relations between the best approximation and the worst approximation. We show that these relations are equivalence relations if the sets are Chebyshev or uniquely remotal. We obtain cosets sets of best approximation and cosets sets of worst approximation. We obtain some results on these sets, for example, compactness and weakly compactness. Finally, we consider the semi-inner products (Lumer-Giles) and semi-inner(usual).

Chebyshev sets, Uniquely remotal sets, Cosets best aprrpximation sets, Cosets worst approximation sets, Equivalence relations