Real renormings on complex Banach spaces (Q942934)

The authors prove that every complex Banach space admits an equivalent real norm that is far away from being a complex norm. Furthermore, this real norm can be chosen to share many properties with complex norms, but it is still not a complex norm. More precisely, by considering norm-attaining functionals, they show that any real Banach space of dimension larger than \(1\) can be equivalently renormed to have a convex subset in its unit ball with non-empty interior relative to the unit sphere. Furthermore, following an algebraic approach, they prove that any complex Banach space admits an equivalent real norm not having a subset as in the above, but still not coming from a complex norm.