Every symmetric, nonnegative definite bilinear functional f satisfies the inequality
(f (x, y))2 ≤ f (x, x)f (y, y)
for every x, y ∊ X.
A symmetric, positive definite bilinear functional on a linear space X is called an inner product. It is customary to use a special notation to denote the inner product. We will use xTy to denote f (x, y) when f is an inner product. By definition, an inner product satisfies the following properties for every x, x1, x2, y ∊ X: