This equation defines the desired fixed point mapping, where each of the m2 matrix elements on the right side of Eq. (8.14) are interpreted as MISO problems. Proof of the fact that the design of each MISO system yields a satisfactory MIMO design is based on the Schauder fixed point theorem.7 The theorem defines a mapping Y(Td) where each member of Td is from the acceptable set ℑd (see Fig. 5.10). If this mapping has a fixed point, i.e., Td ∈ ℑd, then this Td is a solution of Eq. (8.14).