A multilinear generalisation of the Cauchy-Schwarz inequality

We prove a multilinear inequality which in the bilinear case reduces to the Cauchy-Schwarz inequality. The inequality is combinatorial in nature and is closely related to one established by Katz and Tao in their work on dimensions of Kakeya sets. Although the inequality is "elementary" in essence, the proof given is genuinely analytical insofar as limiting procedures are employed. Extensive remarks are made to place the inequality in context.