# Cauchy-Schwarz Inequality

The **Cauchy-Schwarz Inequality** (which is known by other names, including **Cauchy's Inequality**, **Schwarz's Inequality**, and the **Cauchy-Bunyakovsky-Schwarz Inequality**) is a well-known inequality with many elegant applications.

## Contents

## Elementary Form

For any real numbers and ,

,

with equality when there exist constants not both zero such that for all , .

### Proof

There are several proofs; we will present an elegant one that does not generalize.

Consider the vectors and . If is the angle formed by and , then the left-hand side of the inequality is equal to the square of the dot product of and , or . The right hand side of the inequality is equal to . The inequality then follows from , with equality when one of is a multiple of the other, as desired.

### Complex Form

The inequality sometimes appears in the following form.

Let and be complex numbers. Then

.

This appears to be more powerful, but it follows immediately from

.

## General Form

Let be a vector space, and let be an inner product. Then for any ,

,

with equality if and only if there exist constants not both zero such that .

### Proof 1

Consider the polynomial of

.

This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., must be less than or equal to , with equality when or when there exists some scalar such that , as desired.

### Proof 2

We consider

.

Since this is always greater than or equal to zero, we have

.

Now, if either or is equal to , then . Otherwise, we may normalize so that , and we have

,

with equality when and may be scaled to each other, as desired.

### Examples

The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the **Cauchy-Schwarz Inequality for Integrals**: for integrable functions ,

,

with equality when there exist constants not both equal to zero such that for ,

.

## Problems

### Introductory

- Consider the function , where is a positive integer. Show that .Source

### Intermediate

### Olympiad

is a point inside a given triangle . are the feet of the perpendiculars from to the lines , respectively. Find all for which

is least.

## Other Resources

### Books

- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.