is an inner product space over with the inner product
An inner product space over is usually called a Euclidean space.
The following properties of an inner product can be deduced from the four axioms in Definition 1:
- for all if and only if , and
An important property shared by all inner products is the Cauchy-Schwarz inequality and, for an inner product space, one of the most useful inequalities in mathematics.
The proof of this can be done in a number of different ways. The most common proof is to consider the quadratic function in
and derive the inequality from the non-positive discriminant. We will first present this proof.
Proof: Let be given. If , the assertion is trivial, so we may assume that . Let and consider
which is a real quadratic polynomial with real coefficients. Because of axiom (1.), we know that for all real , and hence can have no real simple roots. The discriminant of must therefore be non-positive
Since this inequality must hold for any pair of vectors, it must hold if is replaced by , so we also have the inequality
But , so
If , then the statement of the theorem is trivial; if not, then we may divide equation (2) by the quantity to obtain the desired inequality
Because of axiom (1.), can have a real (double) root only if for some . Thus, equality can occur in the discriminant condition in equation (1) if and only if and are linearly dependent.
We will now present a matrix proof, focusing on the complex vector space, which is perhaps the simplest proof of the Cauchy-Schwarz inequality.
Proof: For any vectors we noticed that,
By taking the determinant for the matrix,
the inequality follows at once,
Equality occurs if and only if the matrix has rank 1; that is, and are linearly dependent.
 Zhang, Fuzhen. Matrix theory: basic results and techniques. Springer Science & Business Media, 2011.