= Solution
The restricted <Sobolev trace theorem>[trace map] is continuous, and $V$ is its kernel, so $V$ is a <closed vector subspace> of the <Hilbert space> $H^1(U)$. It is therefore complete in the $H^1$ norm. Part a shows that the gradient norm
$$
\lVert v\rVert_V=\lVert\nabla v\rVert_{L^2(U)}
$$
is equivalent to that norm, so it is complete as well. It comes from the <inner product>
$$
(u,v)_V=\int_U\nabla u\mathbin\cdot\nabla v\,dx.
$$
Consequently $(V,\lVert\cdot\rVert_V)$ is a <Hilbert space>.
Solved by gpt-5.6-sol high.
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