I'm quite new to the Einstein Notation (summation convention) and was wondering if anyone can clear it up for me. The notation says that if an index say $i$ appears $\textbf{twice}$ then we sum up all the values.
I want to prove that
$$\forall u,v,w \in \mathbb{R}^3 \\ (u + v) \cdot w = u \cdot w + v \cdot w $$
Here is my attempt (and this is where my confusion will show).
$$ u \equiv u_{i}e_{i}, v \equiv v_{j}e_{j}, w \equiv w_{k}e_{k} \quad \text{for} \: i,j,k \in \{ 1,2,3 \} \\ u+v = u_{i}e_{i} + v_{j}e_{j} \\ u+v = (u_{c} + v_{c})e_{c}$$
(for c in {1,2,3}). Since I can see that if I expand the summation, I can make the i's and j's with the same indice variable. But th notation specifically states that only if the index appears TWICE. It appears three times here so I'm not sure if this is still to the convention. Can someone confirm?