Advanced Linear Algebra
《Advanced Linear Algebra》 · 第 205 页 · Chapter 9「Real and Complex Inner Product Spaces」
Definition Let $V$ be a vector space over $F=\mathbb R$ or $F=\mathbb C$. An inner product on $V$ is a function $\langle\cdot,\cdot\rangle:V\times V\to F$ with the following properties: 1) (Positive definiteness) For all $v\in V$, $\langle v,v\rangle\ge0$ and $\langle v,v\rangle=0\iff v=0$. 2) For $F=\mathbb C$: (Conjugate symmetry) $\langle u,v\rangle=\overline{\langle v,u\rangle}$. For $F=\mathbb R$: (Symmetry) $\langle u,v\rangle=\langle v,u\rangle$. …