Study on Orthogonal Sets for Birkhoff Orthogonality

We introduce the notion of orthogonal sets for Birkhoff orthogonality, which we will call Birkhoff orthogonal sets in this paper. As a generalization of orthogonal sets in Hilbert spaces, Birkhoff orthogonal sets are not necessarily linearly independent sets in finite-dimensional real normed spaces. We prove that the Birkhoff orthogonal set A={x1,x2,…,xn}(n≥3) containing n−3 right symmetric points is linearly independent in smooth normed spaces. In particular, we obtain similar results in strictly convex normed spaces when n=3 and in both smooth and strictly convex normed spaces when n=4. These obtained results can be applied to the mutually Birkhoff orthogonal sets studied in recently.