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 $\mathbb C$?





$$q(x_1,...,x_n) = x_1^2 + ... + x_n^2$$

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 $n \ge 3$.






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  • $\begingroup$ Thank you. What about fields which are also ordered sets? $\endgroup$ – heiner Nov 24 '09 at 16:21
  • 2
    $\begingroup$ The ordering needs to be compatible with the field operations, but yes, you can do that. $\endgroup$ – Qiaochu Yuan Nov 24 '09 at 16:29
  • $\begingroup$ Just a trivial remark: for the quaternions, which you can think of as a "skew" field, you cannot define bilinear forms, but only sesquibilinear forms. At any rate, the important property is not so much bilinearity as non-degeneracy. $\endgroup$ – José Figueroa-O'Farrill Nov 24 '09 at 16:53
  • $\begingroup$ I feel like there's a way to do this for fields of characteristic zero, or if not, at least real closed number fields and their algebraic closures. $\endgroup$ – Harry Gindi Nov 25 '09 at 0:17


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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:

  1. $\langle x,x\rangle\neq 0$ for all $x\neq 0$, $x\in E$.
  2. $\langle x,y\rangle=\langle y,x\rangle^*$ for all $x,y\in E$.
  3. $x\mapsto \langle x,y\rangle$ 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:

  1. $\langle x,x\rangle\neq0$ whenever $x\neq0$.
  3. $|\langle x,y\rangle|^2\leq|\langle x,x\rangle||\langle y,y\rangle|$.

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

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