We define the Sobolev spaces of functions $W^{m,p}(\Omega)$ where $\Omega$ is an open subset of $\R^n$ $m\ge 0$ is an integer and $p\in[1,+\infty]$

The spaces $W^{0,p}(\Omega)$ are simply defined to be the spaces $L^p(\Omega)$ of Lebesgue $p$ summable functions. We then define the space $W^{m,p}(\Omega)$ to be the space of functions $u\in L^p(\Omega)$ which have weak derivatives $g=(g_1,\ldots,g_n)$ such that $g_i\in W^{m-1,p}(\Omega)$

The space $W^{m,p}$ turns out to be a Banach space when endowed with the norm $$ \Vert u \Vert_{W^{m,p}}= \sum_{k=0}^m \sum_{i_1=1}^n \cdots \sum_{i_k=1}^n \left[\int_\Omega \left|\frac{\partial^k u(x)}{\partial x_{i_1}\cdots\partial x_{i_k}}\right|^p \, dx\right]^{\frac 1 p} $$ i.e. the sum of the $L^p$ norms of $u$ and of all weak derivatives of $u$ up to the $m$ th order.

Of particular interest are the spaces $H^m(\Omega):=W^{m,2}(\Omega)$ which turn out to be Hilbert spaces with the scalar product given by $$ (u,v)_{H^m(\Omega)}=\sum_{k=0}^m \sum_{i_1=1}^n \cdots \sum_{i_k=1}^n \int_\Omega \frac{\partial^k u(x)}{\partial x_{i_1}\cdots\partial x_{i_k}} \frac{\partial^k v(x)}{\partial x_{i_1}\cdots\partial x_{i_k}} \, dx. $$