Cauchy-Schwarz inequality
The
Cauchy-Schwarz inequality, also known as the
Schwarz inequality, or the
Cauchy-Bunyakovski-Schwarz inequality, is a useful inequality encountered in many different settings, such as
linear algebra talking about
vectors, and in
analysis talking about
infinite series and
integration of products. The inequality states that if
x and
y are elements of a
real or
complex inner product spaces then
- |<x, y>|2 ≤ <x, x> · <y, y>
The two sides are equal if and only if
x and
y are
linearly dependent.
An important consequence of the Cauchy-Schwarz inequality is that the inner product is a continuous function.
Formulated for Euclidean space Rn, we get
- ( ∑ xi yi )2 ≤ ( ∑ xi2) · ( ∑ yi2)
In the case of
square-integrable complex-valued
functionss, we get
- | ∫ f * g dx|2 ≤ ( ∫ |f|2 dx) · ( ∫ |g|2 dx)
These latter two are generalized by the Hölder inequality.
See also Triangle inequality.