Finite-dimensional Hopf algebras are Frobenius

By [F2]–[F3], the nonzero $\lambda$ spans ${\int }_{H^{\ast }}^{\mathrm{\ell }}$. Restricting the isomorphism [F5] to this one-dimensional first factor shows that $h\mapsto \lambda \leftharpoonup h$ is an isomorphism $H\to H^{\ast }$. By [F4], this map sends $h$ to the functional $x\mapsto \lambda (x{S}_H(h))$. Since $S_H$ is bijective, $\mathrm{\Theta }:H\to H^{\ast }$, $\mathrm{\Theta }(b)(x)=\lambda (xb)$, is an isomorphism.

Define $v_i:={\mathrm{\Theta }}^{-1}(u_i^{\ast })$. Then $\lambda (u_j{v}_i)=u_i^{\ast }(u_j)={\delta }_{ji}$. If ${\sum }_i{c}_i{v}_i=0$, applying $u_j\mapsto \lambda (u_j\cdot )$ gives $c_j=0$ for every $j$; hence $(v_i)$ is a basis. Writing $a={\sum }_i{d}_i{u}_i$, the identities in [F6] give $\lambda (a{v}_j)=d_j$, so $a={\sum }_i\lambda (a{v}_i)u_i$. Writing instead $a={\sum }_i{e}_i{v}_i$, the dual-pairing identity gives $\lambda (u_{j}a)=e_j$, so $a={\sum }_i\lambda (u_{i}a)v_i$.

The first dual-basis identity in step 2.1 implies that if $\mathrm{\Phi }(a)=0$, then $\lambda (a{v}_i)=0$ for all $i$ and hence $a=0$; thus $\mathrm{\Phi }$ is injective and is surjective because $\dim H=\dim H^{\ast }$. The second identity and the fact that $\mathrm{\Theta }$ is an isomorphism show nondegeneracy in the other variable as well. Finally, $\lambda ((ab)c)=\lambda (a(bc))$ by associativity of multiplication, so the form is associative. Definition [F1] now makes $\lambda$ a Frobenius functional and $H$ a Frobenius algebra.