Consider, you have a problem in an infinite dimensional Hilbert or Banach space (think of a PDE or an optimization problem in such a space) and you have an algorithm that converges weakly to a solution. If you discretize the problem and apply the corresponding discretized algorithm to the problem, then weak convergence is convergence in every coordinate and hence also strong. My question is:
Does this kind of strong convergence feel or look any different from convergence obtained from good old plain strong convergence of the original infinite algorithm?
Or, more concrete:
What kind of bad behaviour can happen with a "discretized weakly convergence method"?
I myself are usually not quite happy when I can only prove weak convergence but up to now I could not observe some problem with the outcome of the methods even if I scale the problem discretized problems to higher dimensions.
Note that I am not interested in the "first discretize than optimize" vs. "first optimize than discretize" problem and I am aware of problems that can occur if you apply an algorithm to a discretized problem that does not share all properties with the problem for which the algorithm was designed for.
Update: As a concrete example consider an optimization problem with a variable in $L^2$ and solving it with something like (an inertial) forward-backward splitting or some other method for which only weak convergence in $L^2$ is known. For the discretized problem you can use the same method and with the correct discretization you get the same algorithm is if you discretized the algorithm directly. What can go wrong when you increase the discretization accuracy?