  Discontinuous Petrov Galerkin (DPG) methods are made easily implementable
using `broken' test spaces, i.e., spaces of functions with no continuity
constraints across mesh element interfaces. Broken spaces derivable from a
standard exact sequence of first order (unbroken) Sobolev spaces are of
particular interest. A characterization of interface spaces that connect the
broken spaces to their unbroken counterparts is provided. Stability of certain
formulations using the broken spaces can be derived from the stability of
analogues that use unbroken spaces. This technique is used to provide a
complete error analysis of DPG methods for Maxwell equations with perfect
electric boundary conditions. The technique also permits considerable
simplifications of previous analyses of DPG methods for other equations.
Reliability and efficiency estimates for an error indicator also follow.
Finally, the equivalence of stability for various formulations of the same
Maxwell problem is proved, including the strong form, the ultraweak form, and a
spectrum of forms in between.
