I have a basic understanding of the Sobolev spaces. Let $\Omega \subset \mathbb{R}^n$ and $k=1,2,3... .$ I understand that $W^{-1,k}(\Omega)$ is defined as the dual of $W_0^{1,k'}(\Omega)$ where $k'$ is the conjugate exponent of $k.$ In other words, $W^{-1,k}(\Omega)$ is a normed linear space which consists of all the functionals $f :W_0^{1,k'}(\Omega) \rightarrow \mathbb{R}$ equipped with operator norm defined by $||f||_{W^{-1,k}(\Omega)}= \sup\limits_{||\phi||_{W^{1,k'}(\Omega)}=1}|f(\phi)|.$ I recently came across the space $W^{-1,k}_{loc}(\Omega).$
I would like to understand the followings:
1. Is there a norm (or atleast a metric) on $W^{1,k}_{loc}(\Omega)$?
2. What is the definition of $W^{-1,k}_{loc}(\Omega)$? How to interpret these spaces? (In view of (1) is it the dual of $W^{1,k}_{loc}(\Omega)$ under the norm/metric )
3. What do we mean by compact sets and bounded sets in $W^{-1,k}_{loc}(\Omega)$?
I would say that all that you wrote for $W^{1,k}(\Omega)$ can be easily seen to be true for $W^{1,k}_{loc}(\Omega)$, where for $W^{1,k}_{loc}(\Omega)$ we mean all the functions in $W^{1,k}(K)$ for any compact set $K \subset \Omega$.
Side remark: a nice (not unique) characterization of an element is its dual is the following. Embed $W^{1,p}$ in $L^p\times (L^p)^n$ and using Hahn-Banach (for the extension) and use then Riesz' representation theorem (for in $L^p$).