What is the Cauchy-Schwarz inequality?

Overview

The Cauchy-Schwarz inequality is also known as the Cauchy-Bunyakovsky-Schwarz inequality, or the Schwarz inequality. We use this inequality to bound expected values that are difficult to calculate.

Forms of the Cauchy-Schwarz inequality

There are different forms of the Cauchy-Schwarz equation. These are:

• Random variance

• Vector form

Let's discuss them in detail.

Random variance

Random variance allows us to split $E[X_1, X_2]$ into their upper bounds in two parts. In this, we assign each part to each random variable.

The formula for this inequality is as follows:

Note: $X$ and $Y$ have finite variances.

This means that for the two random variables $X$ and $Y$, the expected value of the square of them multiplied together $E(XY)^2$ is always less than or equal to the expected value of the product of the squares of each $E(X^2)E(Y^2)$.

Vector form

The vector form, for all the vectors $X$ and $Y$ of real inner product space, is defined as follows:

This proves the following:

This equality only holds if the two vectors are linearly dependent.

For a complex vector space, we define it as follows:

This equality holds if, and only if, $x$ and $y$ are linearly dependent.

Example

Using the Cauchy-Schwarz inequality, let's find the maximum of $x \ + \ 2y \ + \ 3z$, given that $x^2\ + \ y^2\ + \ z^2\ = \ 1$.

Solution

We will replace these values in the Cauchy-Schwarz inequality. We get the following:

The sum of the squares of the whole part gives us $14$.

Thus, the following holds:

This happens when the following equality holds:

As $x^2\ + \ y^2\ + \ z^2\ = \ 1$, by equality we get the following:

This is the maximum value of the equation.

Applications

The Cauchy-Schwarz inequality has a vast number of applications. We use it in the following domains:

• Probability and statistics

• The Hilbert space theory

• Classical real and complex analysis

• Numerical analysis

• Qualitative theory of differential equations