So far our methods for analyzing the measured signal have been based on the idea of matching the data against various potential complex exponential components to see which ones match best. The matching is done using the complex dot product, . In the ideal case this dot product is large, for those values of ? that correspond to an actual component of the signal; otherwise it is small. Why this should be the case is the Cauchy-Schwarz inequality (or sometimes, depending on the context, just Cauchy s inequality, just Schwarz s inequality, or, in the Russian literature, Bunyakovsky s inequality). The proof of Cauchy s inequality rests on four basic properties of the complex dot product. These properties can then be used to obtain the more general notion of an inner product.