Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.

Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.

Statement of the inequality

The Cauchy–Schwarz inequality states that for all vectors

u

{\displaystyle \mathbf {u} }

and

v

{\displaystyle \mathbf {v} }

of an inner product space

where

⟨

⋅

,

⋅

⟩

{\displaystyle \langle \cdot ,\cdot \rangle }

is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean

ℓ

2

{\displaystyle \ell _{2}}

norm, called the canonical or induced norm, where the norm of a vector

u

{\displaystyle \mathbf {u} }

is denoted and defined by

‖

u

‖

:=

⟨

u

,

u

⟩

,

{\displaystyle \|\mathbf {u} \|:={\sqrt {\langle \mathbf {u} ,\mathbf {u} \rangle }},}

where

⟨

u

,

u

⟩

{\displaystyle \langle \mathbf {u} ,\mathbf {u} \rangle }

is always a non-negative real number (even if the inner product is complex-valued).

By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:

Moreover, the two sides are equal if and only if

u

{\displaystyle \mathbf {u} }

and

v

{\displaystyle \mathbf {v} }

are linearly dependent.