Inner product/K/Cauchy-Schwarz/Fact

Let ${\displaystyle {}V}$ denote a vector space over ${\displaystyle {}{\mathbb {K} }}$, endowed with an inner product ${\displaystyle {}\left\langle -,-\right\rangle }$. Let ${\displaystyle {}\Vert {-}\Vert }$ denote the associated norm.

Then the Cauchy-Schwarz estimate holds, that is,

${\displaystyle {}\vert {\left\langle v,w\right\rangle }\vert \leq \Vert {v}\Vert \cdot \Vert {w}\Vert \,}$

for all

${\displaystyle {}v,w\in V}$.