## 2017年12月4日月曜日

### 数学 - 線型代数 - 複素数、複素ベクトル空間 - 複素平面(絶対値、等号、不等号、共役、積、商)

1. $\begin{array}{}{\left|1-\stackrel{-}{\alpha }\beta \right|}^{2}-{\left|\alpha -\beta \right|}^{2}\\ =\left(1-\stackrel{-}{\alpha }\beta \right)\stackrel{-}{\left(1-\stackrel{-}{\alpha }\beta \right)}-\left(\alpha -\beta \right)\stackrel{-}{\left(\alpha -\beta \right)}\\ =\left(1-\stackrel{-}{\alpha }\beta \right)\left(1-\alpha \stackrel{-}{\beta }\right)-\left(\alpha -\beta \right)\left(\stackrel{-}{\alpha }-\stackrel{-}{\beta }\right)\\ =1+{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}-\stackrel{-}{\alpha }\beta -\alpha \stackrel{-}{\beta }-{\left|\alpha \right|}^{2}-{\left|\beta \right|}^{2}+\alpha \stackrel{-}{\beta }+\stackrel{-}{\alpha }\beta \\ =1+{\left|\alpha \right|}^{2}{\left|\beta \right|}^{2}-{\left|\alpha \right|}^{2}-{\left|\beta \right|}^{2}\\ =\left(1-{\left|\alpha \right|}^{2}\right)\left(1-{\left|\beta \right|}^{2}\right)\\ >\left(1-1\right)\left(1-1\right)\\ =0\end{array}$

よって、

$\begin{array}{}{\left|1-\stackrel{-}{\alpha }\beta \right|}^{2}-{\left|\alpha -\beta \right|}^{2}>0\\ {\left|\alpha -\beta \right|}^{2}<{\left|1-\stackrel{-}{\alpha }\beta \right|}^{2}\\ \left|\alpha -\beta \right|<\left|1-\stackrel{-}{\alpha }\beta \right|\\ \frac{\left|\alpha -\beta \right|}{\left|1-\stackrel{-}{\alpha }\beta \right|}<1\\ \left|\frac{\alpha -\beta }{1-\stackrel{-}{\alpha }\beta }\right|<1\end{array}$