  We introduce a very weak space-time variational formulation for the wave
equation, prove its well-posedness (even in the case of minimal regularity) and
optimal inf-sup stability. Then, we introduce a tensor product-style space-time
Petrov-Galerkin discretization with optimal discrete inf-sup stability,
obtained by a non-standard definition of the trial space. As a consequence, the
numerical approximation error is equal to the residual, which is particularly
useful for a posteriori error estimation.
  For the arising {discrete linear systems} in space and time, we introduce
efficient numerical solvers that appropriately exploit the equation structure,
either at the preconditioning level or in the approximation phase by using a
tailored Galerkin projection. This Galerkin method shows competitive behavior
concerning {wall-clock} time, accuracy and memory as compared with a standard
time-stepping method in particular in low regularity cases. Numerical
experiments with a 3D (in space) wave equation illustrate our findings.
