University of Oxford Mathematician Dr Tom Crawford introduces the concept of a Bilinear Form, Inner Product, Sesquilinear Form and Inner Product Space.

The video begins with the definition of a Bilinear Form with a concrete example of the dot product on R^n. This is shown to also satisfy the criteria to be symmetric and positive definite, thus making it an Inner Product. The concept of an Inner Product Space is then introduced as a Real Vector Space equipped with an Inner Product. A second example involving an integral over the space of real polynomials is then explored. In the second part of the video Orthonormal Sets are introduced via a definition and then the proof of a lemma stating that any Orthonormal Set in an Inner Product Space is Linearly Independent. The video concludes with a final definition of a Sesquilinear Form and a discussion of a Complex Inner Product Space.