Understanding of multi-index notation - specific example I'm having trouble to really come to terms on how to understand this multi-index notation. 

Given $x=(x_1, \dotsc, x_n) \in K^n$ and $\alpha \in \mathbb{Z}^n$, we write $x^{\alpha} = \prod_{i=1}^n {x_i}^{\alpha_i}$. We consider the multiplicative seminorm, for $\rho \in \mathbb{R}^n \cup \{ - \infty \}$,

$$\delta(\rho) : K[x_1, \dotsc x_n] \rightarrow \mathbb{R}_{\geq 0}, \\
 \sum_{\alpha} c_\alpha x^\alpha \mapsto \max_{\alpha} \{ |c_\alpha| \cdot \exp(\sum_{i=1}^n \rho_i \alpha_i) \}$$
where $|.|$ denotes the absolute value on $K$.

I tried to better understand this by setting $n=3$. This should give me, for $\alpha=(\alpha_1, \alpha_2, \alpha_3)$, $\sum_{\alpha} c_\alpha x^\alpha = c_{\alpha_1} x^{\alpha_1} + c_{\alpha_2} x^{\alpha_3} + c_{\alpha_3} x^{\alpha_3}= c_{\alpha_1} \prod_{i=1}^3 {x_i}^{\alpha_{i1}} + c_{\alpha_2} \prod_{i=1}^3 {x_i}^{\alpha_{i2}} + c_{\alpha_3} \prod_{i=1}^3 {x_i}^{\alpha_{i3}}$, with each $c_{\alpha_i} \in K$. BUT: Logically, there are no $\alpha_{i1}$ etc.??? So maybe I missunderstand the index $\alpha$ below the sum? (Similarly, I thought that the $\max$ on the RHS then runs over the coordinates of the $\alpha \in \mathbb{Z}^n$...is this right?)

It would be also great if someone could show me how a sum and a product of two sums of the form $\sum_{\alpha} c_\alpha x^\alpha$ would have to look like! (I'm working on actually proving that the map above is a multiplicative semi-norm, which doesn't work out yet, probably due to a mistake in the understanding of the index...)

Thank you very much!
If $\alpha$ is an $n$-dimensional vector all of whose coordinates are  integers, then $z^\alpha$ means the product $z_1^{\alpha_1}\cdots z_n^{\alpha_n}$; the notation $|\alpha|$ means $\alpha_1+\cdots+\alpha_n$. In this multi-index notation, a multivariable series can be written in the form $\sum_{\alpha}c_{\alpha}z^{\alpha}$ as abbreviation for $\sum_{\alpha_1\in\mathbb{Z}}\cdots\sum_{a_n\in\mathbb{Z}} c_{\alpha_1\ldots\alpha_n}z^{\alpha_1}\cdots z^{\alpha_n}$.


  In the three-dimensional case we have
  \begin{align*}
\sum_{\alpha}c_{\alpha}x^{\alpha}
=\sum_{\alpha\in\mathbb{Z}^3}c_{\alpha}\prod_{j=1}^3x_j^{\alpha_j}
=\sum_{\alpha_1\in\mathbb{Z}}\sum_{\alpha_2\in\mathbb{Z}}\sum_{\alpha_3\in\mathbb{Z}}c_{\alpha_1\alpha_2\alpha_3}x_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3}
\end{align*}
For example, you have the function":

$f(x,y,z)=x^2yz^3 $

With $\alpha=(1,0,2)$:

$\partial^\alpha f=\frac{\partial}{\partial x}\frac{\partial^2}{\partial z^2}f=6xyz$