  In this paper, we consider a variational formulation for the Dirichlet
problem of the wave equation with zero boundary and initial conditions, where
we use integration by parts in space and time. To prove unique solvability in a
subspace of $H^1(Q$) with $Q$ being the space-time domain, the classical
assumption is to consider the right-hand side $f$ in $L^2(Q)$. Here, we analyze
a generalized setting of this variational formulation, which allows us to prove
unique solvability also for $f$ being in the dual space of the test space,
i.e., the solution operator is an isomorphism between the ansatz space and the
dual of the test space. This new approach is based on a suitable extension of
the ansatz space to include the information of the differential operator of the
wave equation at the initial time $t=0$. These results are of utmost importance
for the formulation and numerical analysis of unconditionally stable space-time
finite element methods, and for the numerical analysis of boundary element
methods to overcome the well-known norm gap in the analysis of boundary
integral operators.
