Hermitian form over a division ring
Let $D$ be a division ring admitting an involution^{} (http://planetmath.org/Involution2) $*$. Let $V$ be a vector space^{} over $D$. A Hermitian form^{} over $D$ is a function from $V\times V$ to $D$, denoted by $(\cdot ,\cdot )$ with the following properties, for any $v,w\in V$ and $d\in D$:

1.
$(\cdot ,\cdot )$ is additive in each of its arguments,

2.
$(du,v)=d(u,v)$,

3.
$(u,dv)=(u,v){d}^{*}$,

4.
$(u,v)={(v,u)}^{*}$.
Note that if the Hermitian form $(\cdot ,\cdot )$ is nontrivial and if $*$ is the identity on $D$, then $D$ is a field and $(\cdot ,\cdot )$ is just a symmetric bilinear form^{}.
If we replace the last condition by $(u,v)={(v,u)}^{*}$, then $(\cdot ,\cdot )$ over $D$ is called a skew Hermitian form.
Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.
Title  Hermitian form over a division ring 

Canonical name  HermitianFormOverADivisionRing 
Date of creation  20130322 15:41:04 
Last modified on  20130322 15:41:04 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  12 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 15A63 
Defines  Hermitian form 
Defines  skew Hermitian form 