Sesquilinear Forms and Unitary Geometry

In this chapter we generalize the notion of a bilinear form and introduce the concept of a sesquilinear form. In the first section of this chapter we develop some of the basic properties of sesquilinear forms and, in analogy with bilinear forms, introduce the notion of a reflexive sesquilinear form. Examples are Hermitian and skew Hermitian forms. We then prove that a reflexive sesquilinear form is equivalent to a Hermitian or skew Hermitian form. The second section is devoted to the structure of a unitary space, that is, a vector space equipped with a Hermitian or skew-Hermitian form. In our main result we prove Witt’s theorem for a non-degenerate unitary space.