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 real Banach spaces.

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 ture under complex spaces setting with some natural modifications. 


Of course, it is natural to assume that the field is real if the problems come from physics. However, this setting prevents us applying the powerful tools like spectral theory, H^\infty-functional calculus, analytic semigroup theory and so on. 


Is there any major theorem in (Linear or nonlinear) Functional analysis only that works in real Banach space and has no similar theorem under complex spaces settings. Of course, any theorem about "partial order" doesn't account. Please give the ones from the theory of optimization and caclulus of variations. Is there any good reference on such topic?