The notation comes from Evans Partial Differential Equations.
From Appendix A, we are given information about multiindex notation. Assume $ u : U \rightarrow R$, $ x \in U$.
(a) A vector of the form $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)$ where each component $\alpha_i$ is a nonnegative integer, is called a multiindex of order $$|\alpha| = \alpha_1 +\alpha_2 + \ldots \alpha_n.$$
(b) Given a multiindex $\alpha$, define $$ D^\alpha u(x) := \frac{ \partial^{|\alpha|}u(x) }{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n} } = \partial_{x_1}^{\alpha_1} \cdots \partial_{x_n}^{\alpha_n} u. $$
Now the definition of this particular Sobolev space ($W^{1,p}$) is that it is the space of functions such that if $u \in W^{1,p}$, then its norm: $$ ||u||_{W^{1,p}(U)} := \left( \sum_{|\alpha| \leq 1} \int_U |D^\alpha u |^p \;dx \right)^{\frac{1}{p}}$$ is bounded.
So here is my issue: Showing and understanding explicitly that: $$||u||_{W^{1,p}(U)} = ||u||_{L^p(U)} + ||Du||_{L^p(U)}.$$
So my first step in trying to understand how we get from one to the other was to write out all of the multiindices of order 1. We have that there are $p$ such multiindices of the form: $$ \alpha_1 = (1, 0, \ldots, 0)$$ $$\alpha_2 = (0, 1, \ldots, 0) $$ $$\vdots$$ $$ \alpha_n = (0, 0, \ldots, 1).$$
There is only one such alpha when the order is $0$, given by $$ \alpha_0 = (0 , 0 , \ldots, 0)$$.
Now I see that $\alpha_0$ will get me the $||u||_{L^p(U)} $ term, that is fairly obvious. But I am struggling to see how $\alpha_1$ through $\alpha_n$ cause us to arrive at $||Du||_{L^p(U)}$.
Essentially, I cannot see why you can go from line 1 to 2:
$\begin{align} \left( \sum_{|\alpha| = 1} \int_U |D^\alpha u |^p \;dx \right)^{\frac{1}{p}} & = \left( \int_U |D^{\alpha_1} u |^p \;dx + \int_U |D^{\alpha_2} u |^p \;dx + \cdots + \int_U |D^{\alpha_n} u |^p \;dx\right)^{1/p} \;\;\; \textrm(1) \\ & = \left( \int_U |D u |^p \;dx \right)^{\frac{1}{p}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \hfill \textrm(2) \\ &= ||Du||_{L^p(U)} \end{align}$
I have thought about this for a while and perhaps I am overthinking the concept of multiindices.