I was reading Rudin's book on elementary Hilbert space theory. Right off the bat he restricts value of an inner product to being a complex number. I just want some confirmation as to if all the theorems (such as Parseval's identity) holds if we were to restricts value of an inner product to being a real number, or any other field for that matter, (for example, finite field).
2026-04-03 01:26:33.1775179593
Hilbert Theory: validity in fields other than complex
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Most of the theorems are true for real scalars also. A notable exception is the following: if $T$ is a bounded operator on a complex Hilbert space such that $ \langle Tx, x \rangle =0$ for all $x$ then $T$ is the zero operator. This is false for real scalars: just consider rotation by $90$ degrees in the plane. As pointed out in the comments there are other exceptions too when you go deeper into the theory. There are books on FA that do not assume that the scalar field is $\mathbb C$.