I recently rediscovered a text written by one of my professors from Purdue. It has a section on vector spaces. Unfortunately the text clarifies as much as it obfuscates.

So here is my question. Say we have two Hilbert spaces S and S', and a function $\displaystyle \Omega : S \to S'$. $\displaystyle \Omega$ is linear if

$\displaystyle \Omega ( \phi + \psi ) = \Omega \phi +' \Omega \psi$

and

$\displaystyle \Omega ( a \psi ) = a \Omega \psi$

where all $\displaystyle \phi$ and $\displaystyle \psi$ in S and all complex numbers a.

Now a Hilbert space is a metric vector space, right? So $\displaystyle \phi$ and $\displaystyle \psi$ are vectors in S, + is the binary operation of addition as defined in S, and +' is the binary operation of addition in S'. But why is the scalar field defined as the complex numbers? Or is this another one of the text's "simplifications" for Physicists? (I can't think of a problem in Physics where the scalar field is not the complex numbers, but that doesn't mean it must be in general.)

-Dan

Edit: He also defines the metric in terms of complex numbers as well if that makes any difference:

$\displaystyle d: S \times S \to \mathbb{C}$

Now that I think of it, isn't a metric function defined in terms of a real number?