Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality, also known as the Cauchy-Bunyakovsky-Schwarz Inequality or simply the Cauchy Inequality, is a fundamental concept in mathematics that has applications in various branches of mathematics, including linear algebra, analysis, and probability theory.

Statement

The Cauchy-Schwarz Inequality states that for any two vectors $\vec{u}$ and $\vec{v}$ in an inner product space, we have:

$|\langle \vec{u}, \vec{v} \rangle| \leq \|\vec{u}\| \cdot \|\vec{v}\|$

where $\langle \vec{u}, \vec{v} \rangle$ denotes the inner product of $\vec{u}$ and $\vec{v}$, and $\|\vec{u}\|$ and $\|\vec{v}\|$ denote the norms of $\vec{u}$ and $\vec{v}$, respectively.

Geometric Interpretation

The Cauchy-Schwarz Inequality can be interpreted geometrically in terms of the angle between two vectors. If $\theta$ is the angle between $\vec{u}$ and $\vec{v}$, then the inequality states that:

$|\cos \theta| \leq 1$

which is always true for any angle $\theta$.

Proof

There are several ways to prove the Cauchy-Schwarz Inequality, but one common method is to consider the quadratic function:

$f(t) = \|\vec{u} + t \vec{v}\|^2$

where $t$ is a scalar. Since $\|\vec{x}\|^2 = \langle \vec{x}, \vec{x} \rangle$ for any vector $\vec{x}$, we have:

$f(t) = \langle \vec{u} + t \vec{v}, \vec{u} + t \vec{v} \rangle = \|\vec{u}\|^2 + 2t \langle \vec{u}, \vec{v} \rangle + t^2 \|\vec{v}\|^2$

Since $f(t)$ is non-negative for any $t$, the discriminant of $f(t)$ must be non-positive:

$(2 \langle \vec{u}, \vec{v} \rangle)^2 - 4\|\vec{u}\|^2 \|\vec{v}\|^2 \leq 0$

which simplifies to the Cauchy-Schwarz Inequality.

Applications

The Cauchy-Schwarz Inequality has many important applications in mathematics. For example, it can be used to prove the Triangle Inequality, which states that:

$\|\vec{u} + \vec{v}\| \leq \|\vec{u}\| + \|\vec{v}\|$

It can also be used to prove the AM-GM Inequality, which states that for any positive real numbers $a_1, a_2, \dots, a_n$, we have:

$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$

Moreover, the Cauchy-Schwarz Inequality is a powerful tool in probability theory, where it is used to prove several important results, such as the Schwarz Inequality for Covariance and the Cauchy-Schwarz Master Theorem.

Conclusion

The Cauchy-Schwarz Inequality is a fundamental concept in mathematics with many important applications. It provides a simple yet powerful tool for analyzing the relationships between vectors in an inner product space and has been used extensively in various branches of mathematics.

Links

コーシー＝シュワルツの不等式[JA]