Solution
= Solution
The <Sobolev trace theorem> makes evaluation at zero a continuous linear map $T_0:H^1(0,1)\to\mathbb R$. Hence
$$
H=\ker T_0
$$
is a <closed vector subspace> of the <Hilbert space> $H^1(0,1)$. Every closed vector subspace of a Hilbert space is complete with the restricted inner product, so $H$ is a Hilbert space with the standard $H^1$ inner product.