Dual space of the sobolev spaces.

The dual space of any topological vector space $X$ is the space of all bounded linear forms on $X$. Hence, the dual space of $W^{m,p}(\Omega)$ is the space of all linear $F:W^{m,p}(\Omega)\to \mathbb{R}$ such that $|F(v)|\leq C\|v\|_{W^{m,p}}$ for all $v\in W^{m,p}(\Omega)$.

If you are looking for explicit characterizations:

• In the Hilbert space setting $p=2$, you can of course identify $H^m(\Omega)$ with its dual using the Riesz representation theorem (see also this question).
• For $p=2$, you can also construct Sobolev spaces $H^s$ for any real $s$ using Fourier transforms. Then you can show that the dual of $H^s$ is $H^{-s}$.
• If $p>n$ for $\Omega\subset\mathbb{R}^n$, then $W^{1,p}(\Omega)$ embeds densely into the continuous functions, and hence its dual space contains the space of regular Borel measures (such as the Dirac delta).

For details, I recommend Adams, R.A. and Fournier, J.J.F, Sobolev Spaces, 2nd ed., Academic Press, 2003.