I am having trouble solving the following problem, which is Exercise 2.3.24 of Radford's Hopf Algebras:
Let $C = C_n(k)$, where $n \geq 1$, and let $\{e_{i,j}\}_{1 \leq i,j\leq n}$ be a standard basis for $C$. Denote the dual basis for $C^*$ by $\{E_{i,j}\}_{1 \leq i,j\leq n}$. Show that $E_{i,j}E_{\ell,k} = \delta_{j,\ell} E_{i,k}$, for all $1 \leq i,j,k,\ell \leq n$.
Here, $k$ is a field and $C_n(k)$ is the coalgebra $(C_n(k), \Delta, \varepsilon)$, in which $C_n(k)$ is the free $k$-module over $\{e_{i,j}\}_{1 \leq i,j\leq n}$ and \begin{align*} \Delta(e_{i,j})= \sum_{\ell = 1}^n e_{i,\ell} \otimes e_{\ell, j}, \ \varepsilon(e_{i,j}) = \delta_{i,j} \end{align*} are the comultiplication and counity, respectively.
If I understand things correctly, $C^*$ admits an algebra structure. Hence, the product $E_{i,j}E_{\ell,k}$ should be regarding this algebra multiplication (as a composition of these functionals does not make sense). I thought that I could write the $E_{i,j}$ as matrices and compute the multiplications manually, but it seems a little off.
If someone could point out how do I start to solve this question, I would be deeply tankful.
As you noted, in general any comultiplication $\Delta$ on $C$ induces a multiplication on $C^*$ by $$(F\cdot G)(x) = \sum F(x_{(1)})\cdot G(x_{(2)})$$ where $\Delta(x)=\sum x_{(1)}\otimes x_{(2)}$.
Applying this here yields $$(F\cdot G)(e_{u,v}) = \sum_w F(e_{u,w})\cdot G(e_{w,v})\,.$$ So just plug in $F=E_{i,j}$ and $G=E_{\ell, k}$.