Real euclidean spaces Real euclidean spaces have definitions of inner product and norm. Examples in (\mathbb R^n): The usual inner product The unit-radius circumference when considering an unusual inner product Cauchy-Schwarz inequality Let (V) be a real vector space. A form or real function [ \begin{aligned} \langle\cdot,\cdot\rangle\colon V\times V &\rightarrow \mathbb R \ (x, y) &\mapsto \langle x,y\rangle \end{aligned}\ ] is said to be an inner product if, for all (x, y, z \in V) and all (\alpha \in \mathbb R),