Definition of inner product for vector spaces over arbitrary fields Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
 A: Several approaches have been tried to conceive a well-behaved "inner product" on non-Archimedean valued fields. Let's see some examples:
Option 1:
    Let $\lambda\mapsto\lambda^*$ be a field automorphism of order $2$ defined on $K$. Let $E$ be a $K$-vector space. An inner product is a map $\langle,\rangle:E\times E\to K$ such that:


*

*$\langle x,x\rangle\neq 0$ for all $x\neq 0$, $x\in E$.

*$\langle x,y\rangle=\langle y,x\rangle^*$ for all $x,y\in E$.

*$x\mapsto \langle x,y\rangle$ is linear for each $y\in E$.


Note that if axioms 1,2,3 are assumed in the complex case then either 
$\langle ,\rangle$ or $-\langle ,\rangle$ is positive definite.
It is proved in this book1 that $\langle ,\rangle$ satisfies the Cauchy-Schwartz inequality, it induces a norm, $\langle ,\rangle$ is continuous in the topology induced by the norm and more. The down side is that there is no infinite-dimensional Hilbert-like spaces over $K$.
Option 2: A similar approach was taken in the paper: A non-Archimedean inner product, L. Narici, E. Beckenstein, Contemporary Mathematics, AMS, vol.384, pp.187--202 (2005). There, the "inner product" $\langle ,\rangle$ is defined by the axioms:


*

*$\langle x,x\rangle\neq0$ whenever $x\neq0$.

*$x\mapsto \langle x,y\rangle$ is linear for each $y\in E$.

*$|\langle x,y\rangle|^2\leq|\langle x,x\rangle||\langle y,y\rangle|$.


This map defines the norm $\|x\|=|\langle x,x\rangle|^{1/2}$.
Also, it is proved that in $C_0$, the sup norm is induced by $\langle x,y\rangle=\sum x_ny_n$
(for $x=(x_n)_n$ and $y=(y_n)_n$ in $C_0$) if and only if the residue class field of $K$ is real closed. 
Subsequent study of operators in $C_0$ over an ordered non-Archimedean valued field was developed in here2, here3, here4 and here5.
As in the option 1, in this case there are no infinite-dimensional Hilbert-like spaces. This is shown in this other paper6. However, when the valuations take values in ordered groups rather than the real numbers, the situation changes as we see in the following.  
Option 3: It is possible to conceive Quadratic spaces over Krull valued fields. Under certain conditions, it is even possible to have infinite-dimensional Hilbert-like spaces.
For this topic, in particular, I recommend the papers: 
a. On a class of orthomodular quadratic spaces, H. Gross, U.M. Künzi - Enseign. Math, 1985.
b. Banach spaces over fields with a infinite rank valuation - [H.Ochsenius A., W.H.Schikhof] - 1999
c. After that see: Norm Hilbert spaces over Krull valued fields - [H. Ochsenius, W.H. Schikhof] - Indagationes Mathematicae, Elsevier - 2006

1Perez-Garcia, C.; Schikhof, W. H. Locally convex spaces over non-Archimedean valued fields
2J. Aguayo and M. Nova: Non-Archimedean Hilbert like spaces, Bull. Belg. Math. Soc. Simon Stevin, Volume 14, Number 5 (2007), 787-797.
3 José Aguayo, Miguel Nova, and Khodr Shamseddine: Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field, Journal of Mathematical Physics 54, 023503 (2013); https://doi.org/10.1063/1.4789541
4 José Aguayo, Miguel Nova, and Khodr Shamseddine: Inner product on $B^*$-algebras of operators on a free Banach space over the Levi-Civita field, Indagationes Mathematicae, Volume 26, Issue 1, January 2015, Pages 191-205; https://doi.org/10.1016/j.indag.2014.09.006
5 José Aguayo, Miguel Nova, and Khodr Shamseddine: Positive operators on a free Banach space over the complex Levi-Civita field, p-Adic Numbers, Ultrametric Analysis and Applications, April 2017, Volume 9, Issue 2, pp 122–137; https://doi.org/10.1134/S2070046617020029
6 M. P. Solèr: Characterization of hilbert spaces by orthomodular spaces, Communications in Algebra, Volume 23, 1995 - Issue 1,  Pages 219-243; https://doi.org/10.1080/00927879508825218 
A: To get an analogue of hermitian inner products, generalizing the inner product of complex vector spaces, one usually considers fields endowed with an involution (just as complex conjugation) and uses that in the more or less obvious way. This is how one can define the unitary groups, for example; see Dieudonné's Sur les groupes classiques.
A: As Qiaochu wrote, the answer to the question is "not really, but..." Let me amplify on the "but" part.
Positive-definiteness inherently requires an ordering on your field.  Conversely, if you have an ordered field, then the theory of inner products goes through verbatim.  (The ordering does not have to be Archimedean, so this indeed gives lots more examples.)
Let's assume that you have a field K of characteristic different from 2 which cannot be ordered: by a theorem of Artin-Schreier, this is equivalent to -1 being a sum of squares in the field.  Then you don't have "positive definiteness".  What is more, the "standard inner product"
$$q(x_1,...,x_n) = x_1^2 + ... + x_n^2$$
will be isotropic for sufficiently large $n$, i.e., there will exist nonzero vectors
$v = (x_1,...,x_n)$ such that $q(v) = 0$.  For instance, if K is finite, this occurs as soon as $n \ge 3$.
Let me remark that "isotropic inner products" are not inherently worthless. I have a preliminary version of a wonderful book, "Linear Algebra Methods in Combinatorics" by Laszlo Babai, which indeed makes nice use of the above inner product over finite fields, even in characteristic 2.
(See http://www.cs.uchicago.edu/research/publications/combinatorics.  Unfortunately it seems that the book never came to fruition.  I got my copy more than 10 years ago when I took an undergraduate course in combinatorics from Babai.)
On the other hand, to any quadratic form over a field K of characteristic not 2, you can associate a symmetric bilinear form.  See (among infinitely other references) p. 2 of
http://alpha.math.uga.edu/~pete/quadraticforms.pdf
As above, it is plausible that an algebraic substitute for "inner product space" is "vector space endowed with an anisotropic quadratic form", i.e., a regular quadratic form without nonzero vectors v for which $q(v) = 0$.  Witt discovered that you can do a lot of "geometry" in this case: especially, he defined reflection through the hyperplane determined by any anisotropic vector: see (e.g....) pp. 17-18 of the above reference.  More is true here than is included in my introductory notes: for instance the orthogonal group of an anistropic quadratic form has the "compactness properties" of the standard real orthogonal group O(n) (that is, it contains no nontrivial split subtorus).
A: No.  The axiom that fails is positive-definiteness, which doesn't mean anything for an arbitrary field.  But one can still define symmetric bilinear forms.
