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 ofthe 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:

• non-degenerate,

• 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 $ℝ$).

Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced $⟂$ 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 $ℂ$. 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.

It is not uncommon to see hermitian or Hermitean instead of Hermitian. The name is a tribute to Charles Hermite of the Ecole Polytechnique.