Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
I'm reading the Vol. III of "Nonlinear functional analysis and its applications" by Zeidler. Many theorems contained there assume that underling space is a real $B$-space. I have to check one by one to see whether or not it is true under complex spaces setting with some natural modifications.
Of course, it is natural to assume that the field is real in some cases. However, this setting prevents us applying the powerful tools like spectral theory, $H^\infty $-functional calculus, analytic semi-group theory and so on.
Is there any major theorem in (Linear or nonlinear) Functional analysis that works only in a real Banach space and has no similar result under a complex space setting. Of course, any theorem about "partial order, " doesn't account. Please give the ones from the theory of optimization and calculus of variations. Is there any good reference on this topic?