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 $$\alpha \in \mathbb R$$,