  Consider an exact category in the sense of Quillen. Assume that in this
category every morphism has a kernel and that every kernel is an inflation. In
their seminal 1982 paper, Beilinson, Bernstein and Deligne consider in this
setting a t-structure on the derived category and remark that its heart can be
described as a category of formal quotients. They further point out that the
category of Banach spaces is an example, and that here a similar category of
formal quotients was studied by Waelbroeck already in 1962. In the current
article, we give a direct and rigorous construction of the latter category by
considering first the monomorphism category. Then we localize with respect to a
multiplicative system. Our approach gives rise to a heart-like category not
only for the Banach spaces. In particular, the main results apply to categories
in which the set of all kernel-cokernel pairs does not form an exact structure.
Such categories arise frequently in functional analysis.
