Sweedler notation

Remarks

Because $f$ is bilinear, $\sum_i f(a_i, b_i)$ depends only on $\Delta(c)$ and not on the chosen sum; it equals $\tilde f(\Delta(c))$ for the linear map $\tilde f$ on $C \otimes C$ induced by $f$. Example: the counit law reads $\sum \varepsilon(c_{(1)}) c_{(2)} = c = \sum c_{(1)} \varepsilon(c_{(2)})$. Example: for a group-like element $g$, with $\Delta(g) = g \otimes g$, one may take $g_{(1)} = g_{(2)} = g$. Non-example: $\sum c_{(1)} \otimes c_{(1)}$ is not a valid expression, because it is not bilinear in $(c_{(1)}, c_{(2)})$. For $\Delta(x) = x \otimes 1 + 1 \otimes x$ in $K[x]$ it would give $x \otimes x + 1 \otimes 1$. The same element written as $(x + 1) \otimes 1 + 1 \otimes x - 1 \otimes 1$ would give $(x+1) \otimes (x+1) + 1 \otimes 1 + 1 \otimes 1$, a different result.