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In mathematics, 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 applied to vectors, in analysis applied to infinite series and integration of products, and in probability theory, applied to variances and covariances. The inequality states that if x and y are elements of real or complex inner product spaces then
- <math>|\langle x,y\rangle|^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle.<math>
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.
Another form of the Cauchy-Schwarz inequality is given using the notation of norm, as explained under norms on inner product spaces, as
- <math> |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\, <math>
Proof
To prove this inequality note it is trivial in the case y = 0. Thus we may assume <y, y> is nonzero. Further, let <math> \lambda \in \mathbb{R} <math>. Thus we may let
- <math> 0 \leq \langle x-\lambda y,x-\lambda y \rangle <math>
- Which equals
- <math> = \langle x-\lambda y,x \rangle - \lambda \langle x-y,y \rangle <math>
- We now choose <math> \lambda = \langle x,y \rangle \cdot \|y\|^{-2} <math>
- Plugging in for <math> \lambda <math> we wind up getting
- <math> 0 \leq \|x\| ^2 - \langle x,y \rangle^2 \cdot \|y\|^{-2} <math>
- Which is true <math> \iff <math>
- <math> \langle x,y \rangle^2 \leq \|x\|^2 \|y\|^2. <math>
- Taking the square root gives us
- <math> \big| \langle x,y \rangle \big| \leq \|x\| \|y\| <math> Q.E.D.
Notable special cases
- <math>\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).<math>
- <math>\left|\int f^*(x)g(x)\,dx\right|^2\leq\int \left|f(x)\right|^2\,dx \cdot \int\left|g(x)\right|^2\,dx.<math>
These latter two are generalized by the Hölder inequality.
- A notable strengthening of the basic inequality occurs in dimension n = 3, where a stronger equality holds:
- <math>\langle x,x\rangle \cdot \langle y,y\rangle = |\langle x,y\rangle|^2 + |x \times y|^2.<math>
See also
triangle inequality
inner product space for a proof of the Cauchy-Schwarz inequality.
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