Abstract
Let X be a complex normed space. Based on the right norm derivative , we define a mapping
by
The mapping
has a good response to some geometrical properties of X. For instance, we prove that
for all
if and only if X is an inner product space. In addition, we define a
-orthogonality in X and show that a linear mapping preserving
-orthogonality has to be a scalar multiple of an isometry. A number of challenging problems in the geometry of complex normed spaces are also discussed.
Disclosure statement
No potential conflict of interest was reported by the author(s).