Consider two abstract spaces (e.g., measure spaces, topological spaces, manifolds) with natural spaces of functions defined on each and natural spaces of mappings between the two spaces. The operation of pull-back of functions has two arguments, the function and the mapping. Fixing the mapping, this defines a linear mapping from functions to functions called the composition operator. Fixing the function, this defined a nonlinear mapping from mappings to functions, called the superposition operator. To begin, these two operators are discussed in a little generality. Then the importance of the superposition operator is outlined in a setting of ordinary differential equations with quite general time- and parameter-dependence.

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Last Updated: Tue Jun 18 09:19:59 2024