On some approximation problems in metric linear spaces (Q789645)

In this paper the authors prove some rather straightforward extensions of certain results by \textit{G. Pantelidis} [Math. Ann. 184, 30-48 (1969; Zbl 0175.420)] concerning the problem of characterization of elements of best approximation in metric linear spaces. (The corresponding results in the case of normed linear spaces were given by \textit{I. Singer} [Best approximation in normed Linear spaces by Elements of Linear Subspaces (1970; Zbl 0154.145)].) More precisely, they consider the following problem (Problem of simultaneous characterization of a set of elements of best approximation): Given a metric linear space (E,d), a linear subspaces G of E, an element \(x\in E\backslash \bar G\) and a subset M of G, what are the necessary and sufficient conditions in order that every element \(g_ 0\in M\) be an element of best approximation to x by the element of G? They also give a characterization of elements of \(\epsilon\)-approximation ('good approximation') in real metric linear spaces. In the main, the proofs are quite similar to the ones given by G. Pantelidis and I. Singer in their respective works mentioned above.