Orthogonality and smoothness induced by the norm derivatives (Q2037030)

This paper is concerned with variations on the orthogonality (similar to the classical Birkhoff-James orthogonality) in terms of certain derivative notions involving the norm on a linear space \(X\). In general, these notions of orthogonality are not symmetric, transitive, nor the orthogonal complement a linear space. It is generally known that some form of smoothness is related to the linearity of the orthogonal complement. One of the orthogonalities considered for non-zero vectors \(x,y \in X\) is: \(x\perp_{\rho}y\) if and only if for a unit vector \(f \in X^\ast\) that attains its norm at \(\frac{x}{\|x\|}\), there is a \(g\) with similar properties such that \(f(y)=-g(y)\). As in the classical case, it turns out that \(x\) is \(\rho\)-smooth if and only if the orthogonal complement is a subspace of codimension \(1\).