## 每週問題 April 11, 2016

Let $\mathcal{V}$ be a complex inner product space. Show that two vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathcal{V}$ are orthogonal if and only if

$\Vert \alpha\mathbf{x}+\beta\mathbf{y}\Vert^2=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2$

for all pairs of scalars $\alpha$ and $\beta$.

\begin{aligned} \Vert \alpha\mathbf{x}+\beta\mathbf{y}\Vert^2&= \left\langle \alpha\mathbf{x}+\beta\mathbf{y},\alpha\mathbf{x}+\beta\mathbf{y}\right\rangle\\ &=\left\langle \alpha\mathbf{x},\alpha\mathbf{x}\right\rangle+\left\langle \beta\mathbf{y},\beta\mathbf{y}\right\rangle+\left\langle \alpha\mathbf{x},\beta\mathbf{y}\right\rangle+\left\langle \beta\mathbf{y},\alpha\mathbf{x}\right\rangle\\ &=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2+\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle+\overline{\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle}\\ &=\Vert\alpha\mathbf{x}\Vert^2+\Vert\beta\mathbf{y}\Vert^2+2\hbox{Re}(\overline{\alpha}\beta\left\langle \mathbf{x},\mathbf{y}\right\rangle). \end{aligned}

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