The Birkhoff orthogonality in pre-Hilbert \(C^\ast\)-modules (Q2831916)

If \(x, y\) are elements of a normed linear space \((X, \|\cdot\|)\) over a field \(\mathbb{K}\), then \(x\) is said to be orthogonal to \(y\) in the Birkhoff sense if \(\|x + \lambda y\| \geq \|x\|\) for all \(\lambda\in\mathbb{K}\). In this paper, the author nicely characterizes the Birkhoff orthogonality for elements and finite-dimensional subspaces of a pre-Hilbert \(C^*\)-module in terms of the convex hull of continuous linear functionals. He gives a characterization of smoothness at a point in terms of the Birkhoff orthogonality in some function spaces. Moreover, he presents a new proof of a theorem of \textit{R. Bhatia} and \textit{P. Šemrl} [Linear Algebra Appl. 287, No. 1--3, 77--85 (1999; Zbl 0937.15023)].