A characterization of best simultaneous approximations (Q5906492)

Let \(X\) be a compact Hausdorff space and \(Y\) a normed linear space and let \(C(x,y)\) denote the set of all continuous functions from \(X\) to \(Y\), equipped with a norm. The author takes up the problem of approximating a finite number of functions in \(C(x,y)\) simultaneously by a function in a subspace of \(C(x,y)\). The main theorem here is concerned with a characterization of best approximation introduced. As a consequence of his theorem the author obtains a result on best simultaneous approximation by \textit{S. Tanimoto} [J. Approximation Theory 59, No. 3, 359-361 (1989; Zbl 0697.41014)] involving Chebyshev norm.