Cauchy-Schwarz inequality
for all , with equality exactly when the vectors are parallel.
Proof
Consider for . Expanding,
is a quadratic in that never becomes negative, so its discriminant is non-positive:
Rearranging gives .
Observations
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The inequality holds for all scalar products on any real vector space.
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Equality holds if and only if or for some , that is, when the vectors are parallel.
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The geometric formula is well-defined because Cauchy-Schwarz ensures .
Related
Stated in
- Theorem 2.12 (Cauchy-Schwarz Inequality)§2.2 Scalar Product (Dot Product)