Knowlpedia

Cauchy–Schwarz inequality

In an inner product space, the absolute value of an inner product is at most the product of norms.

Cauchy–Schwarz inequality: In an (V,⟨⋅,⋅⟩)(V,\langle\cdot,\cdot\rangle), for all x,y∈Vx,y\in V,

∣⟨x,y⟩∣≤∥x∥ ∥y∥,|\langle x,y\rangle|\le \|x\|\,\|y\|,

where ∥x∥=⟨x,x⟩\|x\|=\sqrt{\langle x,x\rangle} is the induced . Equality holds if and only if xx and yy are linearly dependent.

Remarks

In a , this bounds the dot product by the product of Euclidean lengths. It is central for understanding and for many norm inequalities in inner product geometry.