Sobolev space

Remarks

Here $L^p(U)$ is the Lp space, and functions equal almost everywhere are identified. Each $W^{k,p}(U)$ is a Banach space, and $H^k(U) = W^{k,2}(U)$ is a Hilbert space with inner product $\sum_{\abs{\alpha} \le k} \int_U D^\alpha u \, D^\alpha v \, dx$. For $k = 0$ one gets $W^{0,p}(U) = L^p(U)$. For $1 \le p < \infty$, the smooth functions in $W^{k,p}(U)$ are dense in it (Meyers–Serrin).

Example: $u(x) = \abs{x}$ lies in $W^{1,p}(-1, 1)$ for all $p$. Indeed $u$ is bounded, and its weak derivative $\sgn x$ is bounded, so both lie in every $L^p(-1, 1)$.

Non-example: $\sgn x$ is bounded, so it lies in every $L^p(-1, 1)$, but it has no weak derivative on $(-1, 1)$. So it is in no space $W^{1,p}(-1, 1)$.