As usual is a field. In this post I will prove a rank-nulity theorem for Frobenius algebras. First we need a lemma also known as the **Riesz representation theorem**.

**Lemma**. Let be a finite dimensional -vector space and let be a non-degenerate bilinear form on If then there exists a unique such that for all

*Proof*. The uniqueness is obvious because is non-degenerate. Now let be a basis for and suppose that is the matrix of with respect to this basis, i.e. the -entry of is for all Let be a vector whose -th coordinate is By the theorem in this post, is invertible because is non-degenerate. So the system of equations has a solution Let be the -th coordinate of So

for all Let Then for any we have

**Notation.** Let be a -vector space, a -vector subspace of and a bilinear form on We let

**Theorem**. Let be a Frobenius -algebra. Let the bilinear form and be as stated in Definition 3 and the theorem in this post, respectively. Then

1)

2)

*Proof*. Part 1) is clear because for all So we only need to prove the second part of the theorem. Define the map by for all and Obviously is a -linear map and by part 1). We are now going to prove that is onto. So let Let be the restriction of to By the above lemma, there exists such that

for all Hence and so is onto. Thus, by the rank-nulity theorem for vector spaces, we have