  We prove results on weak convergence for the alternating split Bregman
algorithm in infinite dimensional Hilbert spaces. We also show convergence of
an approximate split Bregman algorithm, where errors are allowed at each step
of the computation. To be able to treat the infinite dimensional case, our
proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak
convergence of the Douglas-Rachford splitting algorithm and on the relation
between the alternating split Bregman and Douglas-Rachford splitting algorithms
discovered by Setzer. Our motivation for this study is to provide a convergent
algorithm for weighted least gradient problems arising in the hybrid method of
imaging electric conductivity from interior knowledge (obtainable by MRI) of
the magnitude of one current.
