Definition clarification for Sobolev spaces defined by distributions. Here is the definition of a Sobolev space according to my notes.

Let $s \in \mathbb R.$ A temperate distribution $u$ belongs to the Sobolev space $H^s(\mathbb R^n)$ if and only if 
  $$ \mathcal F u \in L_{loc}^2(\mathbb R^n)\quad \text{and  } \mathcal F u\in L^2(\mathbb
R^n, (1+\vert \xi \vert^2)^sd\xi).$$
  By defining the norm 
  $$ \Vert u \Vert_{H^s}^2 := \int \vert\mathcal Fu(\xi)\vert^2 (1 + \vert\xi\vert^2)^s d\xi$$
  we find that $H^s$ is a Banach space.

There are two things that I find confusing. Firstly what does $\mathcal Fu \in L_{loc}^2$ mean ? As defined in my notes 
$$ \mathcal Fu : C_0^\infty(\mathbb R^n) \rightarrow \mathbb K$$
where $ C_0^\infty(\mathbb R^n)$ is the set of smooth functions with compact support, so how can it be in $L_{loc}^2(\mathbb R^n$). Secondly and similarly what does $\mathcal F u (\xi)$ in the definition of $\Vert u \Vert_{H^s}$ mean since $\xi \in \mathbb{R}^n ?$
 A: Generally, a distribution $u \in \mathcal{D}'(\mathbb{R}^n)$ is a linear functional on $C_c^\infty(\mathbb{R}^n)$, the space of compactly supported $C^\infty$ functions.
A function $f \in L^1_{\text{loc}}(\mathbb{R}^n)$ defines a distribution by $\varphi \mapsto \int f(x) \, \varphi(x) \, dx.$ We can therefore identify $L^1_{\text{loc}}(\mathbb{R}^n)$ as a subspace of $\mathcal{D}'(\mathbb{R}^n)$. If $u \in \mathcal{D}'(\mathbb{R}^n)$ and there exists $f \in L^1_{\text{loc}}(\mathbb{R}^n)$ such that $\langle u, \varphi \rangle = \int f(x) \, \varphi(x) \, dx$ then we abuse notation and write $u \in L^1_{\text{loc}}(\mathbb{R}^n)$.
That's what happens here. There exists $f \in L^2_{\text{loc}}(\mathbb{R}^n)$ such that $\langle \mathcal{F}u, \varphi \rangle = \int f(x) \, \varphi(x) \, dx.$ This also answers your second question: $\mathcal{F}u(\xi) := f(\xi).$ Actually, not even this is well-defined since $L^2_{\text{loc}}(\mathbb{R}^n)$ consists of equivalence classes of functions that are equal modulo a null-set. But we can just take any representative; they all work the same when used in integrals.
