sesquilinear forms over general fields
Let $V$ be a vector space^{} over a field $k$. $k$ may be of any characteristic.
1 Sesquilinear Forms
Definition 1.
A function $b\mathrm{:}V\mathrm{\times}V\mathrm{\to}k$ is sesquilinear if it satisfies each of the following:

1.
$b(v,w+u)=b(v,w)+b(v,u)$ and $b(v+u,w)=b(v,w)+b(u,w)$ for all $u,v,w\in V$;

2.
For a given field automorphism $\theta $ of $k$, $b(v,lw)={l}^{\theta}b(v,w)$ and $b(lv,w)=lb(v,w)$ for all $v,w\in V$ and $l\in k$.
Remark 2.
It is possible to apply the field automorphism in the first variable but is more common to do so in the second variable. Also, if $\theta \mathrm{=}\mathrm{1}$ the form is a bilinear form^{}.
Sesquilinear forms^{} are commonly ascribed any combination of the following properties:
 •

•
reflexive^{}, (commonly required to define perpendicular^{});

•
positive definite^{} (this condition requires the fixed field of $\theta $, ${k}_{0}$, be an ordered field, such as the rationals $\mathbb{Q}$ or reals $\mathbb{R}$).
Nondegenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities^{} through the induced $\u27c2$ operation. (See polarity (http://planetmath.org/Polarity2).)
2 Hermitian Forms
If ${\theta}^{2}=1$, it is common to exchange notation at this point and use the same notation of $\overline{l}$ for ${l}^{\theta}$ as is common for complex conjugation – even if $k$ is not $\u2102$. Then $\overline{\overline{l}}=l$.
In this notation, Hermitian forms may be defined by the property
$$b(v,w)=\overline{b(w,v)}.$$ 
Remark 3.
Title  sesquilinear forms over general fields 
Canonical name  SesquilinearFormsOverGeneralFields 
Date of creation  20130322 15:58:17 
Last modified on  20130322 15:58:17 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  11 
Author  Algeboy (12884) 
Entry type  Definition 
Classification  msc 47A07 
Classification  msc 15A63 
Classification  msc 11E39 
Classification  msc 51A05 
Synonym  Hermitian form 
Synonym  Hermitean form 
Related topic  ReflexiveNonDegenerateSesquilinear 
Related topic  NonDegenerate 
Related topic  Polarity2 
Related topic  Projectivity^{} 
Related topic  ProjectiveGeometry 
Related topic  Isometry2 
Related topic  ProjectiveGeometry3 
Related topic  ClassicalGroups 
Defines  sesquilinear form 
Defines  Hermitian form 
Defines  bilinear form 
Defines  Hermitean 