Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that for all vectors and in any inner product space,

It is considered one of the most widely used inequalities in math.

Equality in the Cauchy-Schwarz inequality (a.k.a. ) implies that the two vectors and are linearly dependent. ¹

Example: Euclidean Space

In an Euclidean space where the inner product is defined as the dot product, we have

where is the angle between and .

Taking the square root, and we get the Cauchy-Schwarz inequality:

Further, we can see that the inequality is only equal when , in which case and are collinear (and thus linearly dependent).

Relation with The Triangle Inequality

The triangle inequality is a consequence of the Cauchy-Schwarz inequality: ¹

Taking square roots gives the triangle inequality:

Footnotes

1. Cauchy–Schwarz inequality - Wikipedia ↩ ↩²