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 R^{n}, we get
- ( ∑ x_{i} y_{i} )^{2} ≤ ( ∑ x_{i}^{2}) · ( ∑ y_{i}^{2})
In the case of square-integrable complex-valued functions, we get
- | ∫ f^{ *} g dx|^{2} ≤ ( ∫ |f|^{2} dx) · ( ∫ |g|^{2} dx)
These latter two are generalized by the Hölder inequality.
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