I recently noticed that my go-to reference for stuff related to Hilbert spaces uses the terms "Hilbert space" and "complex Hilbert space" synonymously. Hence the following questions:
- Are there any "basic" theorems that only hold in the complex (or the real) case? By "basic" I perhaps mean that they would be discussed in introductory functional analysis courses.
- In particular, what about spectral theory and functional calculus? What motivates the restriction to the complex case?
- Any general guidelines/rules of thumb to quickly evaluate whether a result only holds for complex/real spaces? E.g. some feature in the statement of the theorem or something to look for in the proof.
Here is my favorite result, which is true in the real case ($\mathbb K = \mathbb R$), but fails in the complex case ($\mathbb K = \mathbb C$): The map $f \colon H \to \mathbb K$, $x \mapsto \|x\|_H^2/2$ is (Fréchet) differentiable.