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 \(\alpha \in \mathbb R\),