In this work, we present a methodology for the construction of the attainable region (AR) and for the synthesis of globally optimal reactor networks employing isothermal, steady-state, plug flow reactors (PFR), continuous-stirred tank reactors (CSTR) as well as mixing. The problem is solved using the novel Infinite DimEnsionAl State-space (IDEAS) approach. Within the IDEAS framework, reactor network synthesis is formulated as an infinite dimensional convex (linear) optimization problem. Pointwise identification of the reactor network's attainable region is shown to be equivalent to the solution of an infinite dimensional linear program. A finite dimensional approximation strategy is presented and illustrated on an example from the literature.