Similarities and differences between real and complex Banach spaces: an overview and recent developments

Abstract

There are numerous cases of discrepancies between results obtained in the setting of real Banach spaces and those obtained in the complex context. This article is a modern exposition of the subtle differences between key results and theories for complex and real Banach spaces and the corresponding linear operators between them. We deeply discuss some aspects of the complexification of real Banach spaces and give several examples showing how drastically different can be the behavior of real Banach spaces versus their complex counterparts.

An orthogonality relation in complex normed spaces based on norm derivatives

“…Inspired by (2.1), we present the following definition in complex normed spaces. For an overview of similarities and differences between real and complex normed spaces see [21]. Definition 2.2.…”