# 问题：

Verify that $(·, ·)$ just defined is an inner product on $C^n$.

## 解答

1.由数学归纳的知识，这里只证明$i=2$的简单情况,

$$\left(\left|v_{1}\right\rangle, \lambda_{1}\left|w_{1}\right\rangle+\lambda_{2}\left|w_{2}\right\rangle\right)=\sum v_{i}^{*}\left(\lambda_{1} w_{1i}+\lambda_{2} w_{2 i}\right).$$

$$\lambda_{1}\left(|v\rangle,\left|w_{1}\right\rangle\right)+\lambda_{2}\left(|v\rangle,\left|w_{2}\right\rangle\right)=\sum v_{i}^{} \lambda_{1} w_{1 i}+\sum v_{i}^{} \lambda_{2} w_{2 i}.$$

2.$$(\left|v\right\rangle,\left|w\right\rangle)=v_{1}^{*} w_{1}+v_{2}^{*} w_{2}+\cdots+v_{n}^{*} w_{n}.$$

$$(|w\rangle,|v\rangle)=w_{1}^{*} v_{1}+w_{2}^{*} v_{2}+\cdots+w_{n}^{*} v_{n};$$

3.$$(\left|v\right\rangle,\left|v\right\rangle)=v_{1}^{*} v_{1}+v_{2}^{*} v_{2}+\cdots+v_{n}^{*} v_{n}=\left|v_{1}\right|^{2}+\left|v_{2}\right|^{2}+\cdots+\left|v_{n}\right|^{2}.$$

#### 参考

[1]www.qtumist.com