The author provides several results on characterization of elements of best approximation by a linear subspace \(G\) in a real metric space \(X\) with translation-invariant metric \(d\). Let \(\emptyset\neq G\subset X\), \(x\in X\setminus G\), \(g_0\in G\). Then \(g_0\) is an element of best approximation for \(x\) if and only if there exists \(f\in X_0^\#\) such that (i) \(\|f\|_d=1\); (ii) \(f(g)=0\) for all \(g\in G\); (iii) \(f(x-g_0)= f(x)=d(x,g_0)\). Here \(X_0^\#= \{f:X\to \mathbb{R}:\|f\|_d< \infty\), \(f(0)=0\), \(f\) subadditive\}, a Lipschitz dual of \(X\). Furthermore, simultaneous characterization of best approximations and elements of \(\varepsilon\)-approximation are also considered.