  The purpose of this paper is to discuss a number of issues that crop up in
the computation of Poisson brackets in field theories. This is specially
important for the canonical approaches to quantization and, in particular, for
loop quantum gravity. We illustrate the main points by working out several
examples. Due attention is paid to relevant analytic issues that are
unavoidable in order to properly understand how computations should be carried
out. Although the functional spaces that we use throughout the paper will
likely have to be modified in order to deal with specific physical theories
such as general relativity, many of the points that we will raise will also be
relevant in that context. The specific example of the mock holonomy-flux
algebra will be considered in some detail and used to draw some conclusions
regarding the loop quantum gravity formalism.
