Sobolev norms and completeness

For an open set $\Omega$, an integer $k\ge0$, and $1\le p\le\infty$, the space $W^{k,p}(\Omega)$ consists of $L^p$ equivalence classes whose weak derivatives through order $k$ also belong to $L^p$. For finite $p$, use the pth root of the sum of all derivative pth powers, including $D^0u=u$. For $p=\infty$, take the maximum of the finitely many essential-supremum derivative norms. Other finite sums define equivalent norms, but a numerical norm calculation must state its convention.

By the highest-order seminorm, we mean $|u|_{W^{k,p}}=(\sum_{|\alpha|=k}\|D^\alpha u\|_p^p)^{1/p}$ for finite p and positive k, using only derivatives of exactly order k. For k equal to one it is the gradient seminorm, with constants in its kernel on each connected component. For higher orders, polynomials of lower degree also lie in this highest-order kernel whenever they belong to the underlying $L^p$ space. Boundary or mean restrictions can remove this kernel, leading later to Poincaré inequalities. Domain size also affects scaling: rescaling a coordinate changes the sizes of a function and its derivative by different powers. Dimensionally scaled norms are sometimes convenient in finite-element analysis, but they must not be silently substituted into an unscaled energy computation.

Take a Cauchy sequence in this full norm. Completeness of $L^p$ supplies a limit $u$ for the functions and a limit $g_\alpha$ for each derivative component. For every compactly supported test, Hölder's inequality lets us pass the weak identity to the limit, since the test and its derivatives belong to the appropriate conjugate spaces on their compact support. Consequently $g_\alpha=D^\alpha u$ as a distribution, and it is represented by an $L^p$ function. The limits are compatible because the derivative relation is closed, not because arbitrary independent limits automatically match.

There are only finitely many derivative components up to the fixed order, so componentwise convergence implies convergence in the Sobolev norm. This proves that $W^{k,p}$ is Banach, including the endpoint cases one and infinity. For $1<p<\infty$, finite products of $L^p$ spaces are reflexive, and the closed derivative graph inherits reflexivity. This additional property often supplies weakly convergent subsequences from bounded families. It is separate from completeness: a complete space need not make every bounded sequence strongly convergent, or even weakly sequentially compact at the endpoint exponents.

The notation $H^k$ means $W^{k,2}$. Its norm comes from summing the $L^2$ inner products of matching derivative components, with complex conjugation in the complex-valued setting. Orthogonal projection and Hilbert-space methods can therefore be applied to approximation problems. A minimizing sequence need not converge strongly just because it is bounded; one often first obtains weak convergence and then uses lower semicontinuity. Knowing which topology a compactness argument produces prevents an unjustified passage through nonlinear expressions.

For finite $p$, define $W^{1,p}_0(\Omega)$ as the closure of $C_c^\infty(\Omega)$ in the full first-order Sobolev norm. It is complete because it is a closed subspace. The subscript zero is not a statement about assigning zeros to arbitrary pointwise boundary representatives. On bounded Lipschitz domains it agrees with the kernel of the trace map, a theorem developed later. On the whole space, compactly supported smooth functions are dense for finite $p$, while on a bounded domain they cannot approximate a function with nonzero trace in the same norm.