I'm beginning the study of coalgebras and the sigma notation using the book called Hopf Algebras of M.E. Sweedler. I'm doing the exercises and I don't know if this ideas are realy clear for me and if what I do is realy what I need do here.
Let $C$ a $\mathbb{k}$-vectorial space $\Delta$ the comultiplication ($\Delta:C \rightarrow C\otimes C $) and $\varepsilon$ the counity ($\varepsilon: C\rightarrow \mathbb{k}$
Exercise: verify the following identity. For any $c\in C$ $$\Sigma_{(c)}\varepsilon(c_{(2)})\otimes\Delta(c_{(1)})=\Delta(c)$$
Here I'm using the Sigma Notation, $\Delta(c)=\Sigma_{(c)}c_{(1)}\otimes c_{(2)}$.
What I did:
We have that each $\varepsilon(c_{(2)})$ is one scalar, say $k_{(2)}$. Furthermore $\Delta(c_{(2)})=\Sigma_{(c_{(2)})}c_{(2)_{(1)}}\otimes c_{(2)_{(2)}}$.Therefore, $$ \Sigma_{(c)}\varepsilon(c_{(2)})\otimes\Delta(c_{(1)})=\Sigma_{(c)}\varepsilon(c_{(2)})\otimes\Sigma_{(c_{(2)})}c_{(2)_{(1)}}\otimes c_{(2)_{(2)}}=\Sigma_{(c)}k_{(2)}\otimes\Sigma_{(c_{(2)})}c_{(2)_{(1)}}\otimes c_{(2)_{(2)}}=^*\Sigma_{(c)}k_{(2)}\otimes\Sigma_{(c_{(2)})}c_{(2)_{(1)}}\otimes c_{(2)_{(2)}}=\Sigma_{(c)}\Sigma_{(c_{(2)})}c_{(2)_{(1)}}\otimes c_{(2)_{(2)}}=\Sigma_{(c)}c_{(1)}\otimes c_{(2)}=\Delta(c) $$
But, after $=^*$ I don't know if each equality makes sense. If makes, why?
We have $$\sum{\epsilon(c_{(1)})\otimes c_{(2)}}=\sum{c_{(1)}\otimes \epsilon(c_{(2)})}=c$$ (This is an axiom.) Since scalar multiplication is commutative, we also have $$\sum{\epsilon(c_{(2)})\otimes c_{(1)}}=c$$ (Really the identity is $\sum{\epsilon(c_{(2)})c_{(1)}}=c$). Thus $$\sum{\epsilon(c_{(2)})\otimes \Delta(c_{(1)})}=\sum{1\otimes \Delta(\epsilon(c_{(2)})c_{(1)})}=1\otimes \Delta(c)$$ (the first equality is true since scalars can be pulled in and out of terms of a tensor product) and the ever-confusing abuse of notation has us calling this $\Delta(c)$.