In this long article, the authors continue their work on exploring the connection between proximinality of subspaces and Birkhoff-James orthogonality in Banach spaces. One misses, though, a reference to I. Singer’s monograph [Best approximation in normed linear spaces by elements of linear subspaces. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0197.38601)]. When a Banach space \(X\) is canonically embedded in its bidual \(X^{\ast\ast}\), \(X\) is precisely the set of weak\(^\ast\)-continuous vectors in \(X^{\ast\ast}\), though the authors continue with a different notation. Theorem 3.2 states that if \(M_f\) denotes the set of norm attaining unit vectors for the weak\(^\ast\) continuous functional \(f \in X^{\ast\ast}\) and \(N_g\) is the null space of the weak\(^\ast\)-continuous functional \(g \in X^{\ast\ast}\), then \(f \perp g\) if and only if \(M_f \cap N_g \neq \emptyset\) .
The paper also connects (Theorem 4.3) best approximation to a finite dimensional subspace \(Y \subset X\) to Hahn-Banach extensions from \(Y^{\bot}\).