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**Definition 1** A vector space over the number field or is called an *inner product space* if it is equipped with an *inner product* satisfying for all and scalar ,

- , if and only if ,
- ,
- , and
- .

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.

**Theorem 1**(Cauchy-Schwarz Inequality) Let be an inner product space. Then for all vectors and in over the field or ,

Equality holds if and only if and are linearly dependent.

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

and hence

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.

**References **

[1] Zhang, Fuzhen. *Matrix theory: basic results and techniques*. Springer Science & Business Media, 2011.

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April 22, 2012 at 5:16 am

The Cauchy-Schwarz Inequality 2: The Schur Complement « Guzman's Mathematics Weblog[…] the previous post, two proofs were given for the Cauchy-Schwarz inequality. We will now consider another […]