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?
Best Answer
There are some differences. For example Bishop-Phelps theorem, which holds only in real Banach spaces. In my opinion, this qualifies as a "major theorem".
MR1749671
Lomonosov, Victor A counterexample to the Bishop-Phelps theorem in complex spaces.
Israel J. Math. 115 (2000), 25–28.
Remark. Your statement "natural to assume that the field is real if the problem comes from physics" is completely wrong.
In fact physicists are MORE interested in the complex field than in the real field. The most fundamental theory of physics, quantum mechanics, describes the state of a system as a vector in a COMPLEX Hilbert space. From the point of view of physics, real numbers are just eigenvalues of Hermitian operators on a complex Hilbert space.