Equip $C_0^\infty(D)$ with the <Dirichlet inner product>
$$
(f,g)_\nabla=\frac1{2\pi}\int_D\nabla f(x)\mathbin\cdot\nabla g(x)\,dx.
$$
The <zero-boundary Sobolev space> $H_0^1(D)$ is its <Hilbert space completion> in the norm $\|f\|_\nabla=(f,f)_\nabla^{1/2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 4 / a / ii / Solution

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The Hilbert space completion of an <inner-product space> $V$ is a <Hilbert space> containing an isometric dense copy of $V$. It can be constructed from <Cauchy sequences> in $V$, identifying two sequences when the norm of their difference tends to zero.

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 Incoming links: Hilbert space completion