A Primer on Sesquilinear Forms

The dot product is an important tool for calculations in Rn. For instance, we can use it to measure distance and angles. However, it doesn’t come from just the vector space structure on Rn – to define it implicitly involves a choice of basis. In Math 67, you may have studied inner products on real vector spaces and learned that this is the general context in which the dot product arises. Now, we will make a more general study of the extra structure implicit in dot products and more generally inner products. This is the subject of bilinear forms. However, not all forms of interest are bilinear. When working with complex vector spaces, one often works with Hermitian forms, which toss in an extra complex conjugation. In order to handle both of these cases at once, we’ll work in the context of sesquilinear forms. For convenience, we’ll assume throughout that our vector spaces are finite dimensional. We first set up the background on field automorphisms.