cauchy schwartz inequality

Topics

• linear algebra
• vector inner product

Cauchy-Schwartz inequality bounds the inner product of two vectors by their lengths. Fundamental in proofs across linear algebra and optimization.

For any vectors $x,y$ in an inner product space $\mathbb{V}$:

$$|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert$$

Where:

• $⟨x,y⟩$ is the inner product
• $\Vert x\Vert =\sqrt{⟨x,x⟩}$ is the vector norm

In ${\mathbb{R}}^n$ with dot product:

$$\left|\sum _{i=1}^n{x}_i{y}_i\right|\le \sqrt{\sum _{i=1}^n{x}_i^2}\sqrt{\sum _{i=1}^n{y}_i^2}$$

Key implications:

• Angle between vectors is well-defined via $\cos \theta =⟨x,y⟩/\Vert x\Vert \Vert y\Vert$
• Basis for triangle inequality in normed spaces

Note

The inequality holds for any two vectors, but equality holds if and only if the vectors are linearly dependent.

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