Inner product/R/Cauchy-Schwarz/Fact

The Cauchy-Schwarz estimate

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

Then the Cauchy-Schwarz estimate holds, that is,

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

for all

${}v,w\in V$.

Write a proof