Why are inner product spaces only defined on $\Bbb R$ or $\Bbb C$? A vector space $V$ makes sense over any field $F$, or even a division ring. So why does adding an inner product suddenly not make sense without taking the $F=\Bbb R$ or $\Bbb C$? What are the primary properties we want in our scalar field that forces it to be one of these two? (In particular I don't see why completeness of the scalar field is necessary in a pre-Hilbert space, but even in a Hilbert space completeness of the vector space does not imply completeness of the scalar field.)


*

*Do any problems arise when taking $F$ to be an involutive field with an absolute value satifying $|x^*|=|x|$ and defining $||x||=\sqrt{|\langle x,x\rangle|}$?

*Or, if we stick to the standard definition $||x||=\sqrt{\langle x,x\rangle}$ (which only makes sense when $F$ has a subfield $K$, identified with a subfield of $\Bbb R$, such that $\langle x,x\rangle\in K$ for all $x\in V$), what problems arise if $F$ is not complete, or at least quadratically complete? (Note that the expression $\sqrt{\langle x,x\rangle}$ is evaluated in $\Bbb R$, not $K$.)
 A: One of the properties of an inner product is positive-definiteness, which requires the field of scalars to contain an ordered sub-field; in particular, finite fields, and fields of finite characteristic will not work, as it is not possible to define an order for them compatible with the field operations.
If we wish the inner product to define a norm via $\|x\| = \sqrt{\langle x,x\rangle}$, and we want this to return a scalar in our field, $F$ needs to contain a Euclidean sub-field (that is, an ordered field in which every non-negative number has a square root).
It is possible to, say, define an inner-product space over the field of all real algebraic numbers, or the field of constructible real numbers, from a purely algebraic point of view, but such inner-product spaces lack nice topological properties (a complete metric).
These topological properties become important when studying function spaces-typically "arbitrary" functions are too bizarre to study in any great detail, so we limit ourselves to collections of "nice" functions (for example, "smooth", or perhaps only "continuous" ones).
