Inequalities are among the most important technical tools in mathematics. The concept of an inequality is at least as old as the concept of number, and yet it is only in relatively modern times that inequalities have been studied in a systematic way. Inequalities often occur implicitly in the statement that a function is convex or that a function of two variables defines a metric.

To begin with we will consider inequalities that involve finite sets of real numbers. Let x_1, x_2, ... , x_N and y_1, y_2, ... , y_N be two sets of real numbers. Then Cauchy's inequality asserts that

and Minkowski's inequality is

Both of these inequalities have important geometrical interpretations. Suppose that x is a vector in ℜ^N with coordinates x₁, x₂, ... , x_N and y is a vector in ℜ^N is coordinates y₁, y₂, ... , y_N. Then the Euclidean norm of x is...