= Solution
The zero-boundary <Sobolev space> is
$$
H_0^1(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^1}.
$$
On it define
$$
(u,v)_{H_0^1}=\int_\Omega\nabla u\cdot\nabla v.
$$
If this quadratic form vanishes, then $\nabla u=0$. The <Poincare inequality> gives
$$
\|u\|_{L^2}\leq C\|\nabla u\|_{L^2}=0,
$$
so $u=0$. It is therefore an inner product, and its norm is equivalent to the usual $H^1$ norm.
A function $u\in H_0^1(\Omega)$ is a <weak solution> when
$$
\int_\Omega\nabla u\cdot\nabla\varphi
=-\int_\Omega f\varphi
\qquad\text{for every }\varphi\in H_0^1(\Omega).
$$
This follows from <integration by parts> and incorporates the homogeneous <Dirichlet boundary condition> through membership in $H_0^1$.
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