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In these notes, a theory for sesquilinear forms on product spaces is developed, with the aim of unifying the treatment of diffusion systems and equation on networks. In the first part, a theoretical framework for sesquilinear forms defined on the direct sum of Hilbert spaces is developed. Conditions for the boundedness, ellipticity and coercivity of the sesquilinear form are proved. A criterion of E.M. Ouhabaz is used in order to prove qualitative properties of the abstract Cauchy problem having as generator the operator associated with the sesquilinear form. In the second part we analyze quantum graphs as a special case of forms on subspaces of the direct sum of Hilbert spaces. First, we set up a framework for handling quantum graphs in the case of infinite networks. Then, the operator associated with such systems is identified and investigated. Finally, we turn our attention to symmetry properties of the associated parabolic problem and we investigate the connection with the physical concept of a gauge symmetry.
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