Publisher Summary This chapter discusses bilinear mappings. The chapter assumes E, F, and G as three topological vector spaces, and a bilinear mapping of E × F into G. This means that, for every xo ∈ E the mappings from F into G are linear. The bilinear map Φ is said to be “separately continuous” if, for all xo, yo, these two linear mappings are continuous. Practically all bilinear mappings considered in analysis are separately continuous. But many of them are not continuous.