Cauchy-Schwarzの不等式

$ \forall\bm u,\bm v\in V:|\braket{\bm u|\bm v}|^2\le\braket{\bm u|\bm u}\braket{\bm v|\bm v}

$ \forall\bm u,\bm v\in V:|\braket{\bm u|\bm v}|\le\lVert\bm u\rVert\lVert\bm v\rVert

$ \forall\bm u,\bm v\in V:|\bm u\cdot\bm v|\le\lVert\bm u\rVert\lVert\bm v\rVert

$ \top

$ \implies\forall t\in\R\forall\bm u,\bm v\in V:

$ 0\le\lVert t\bm u+\braket{\bm u|\bm v}\bm v\rVert^2

$ =t^2\lVert\bm u\rVert^2+2\Re(t\braket{\bm u|\braket{\bm u|\bm v}\bm v})+|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2

$ =t^2\lVert\bm u\rVert^2+2\Re(t|\braket{\bm u|\bm v}|^2)+|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2

$ =t^2\lVert\bm u\rVert^2+2t|\braket{\bm u|\bm v}|^2+|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2

$ \implies\forall t\in\R\forall\bm u,\bm v\in V:

$ \begin{dcases}\lVert\bm u\rVert^2\left(t+\left(\frac{|\braket{\bm u|\bm v}|}{\lVert\bm u\rVert}\right)^2\right)^2+|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2-\frac{|\braket{\bm u|\bm v}|^4}{\lVert\bm u\rVert^2}\ge0&\text{if }\lVert\bm u\rVert\neq0\\0=|\braket{\bm u|\bm v}|\le\lVert\bm u\rVert\lVert\bm v\rVert=0&\text{otherwise}\end{dcases}

$ \implies\forall\bm u,\bm v\in V:

$ \begin{dcases}|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2-\frac{|\braket{\bm u|\bm v}|^4}{\lVert\bm u\rVert^2}\ge0&\text{if }\lVert\bm u\rVert\neq0\\0=|\braket{\bm u|\bm v}|\le\lVert\bm u\rVert\lVert\bm v\rVert&\text{otherwise}\end{dcases}

$ \because\lVert\bm u\rVert^2\left(t+\left(\frac{|\braket{\bm u|\bm v}|}{\lVert\bm u\rVert}\right)^2\right)^2\ge0

$ \implies\forall\bm u,\bm v\in V:

$ \begin{dcases}\lVert\bm v\rVert^2-\frac{|\braket{\bm u|\bm v}|^2}{\lVert\bm u\rVert^2}\ge0&\text{if }\lVert\bm u\rVert\neq0\land\braket{\bm u|\bm v}\neq0\\0=|\braket{\bm u|\bm v}|\le\lVert\bm u\rVert\lVert\bm v\rVert&\text{otherwise}\end{dcases}

$ \underline{\implies\forall\bm u,\bm v\in V:|\braket{\bm u|\bm v}|\le\lVert\bm u\rVert\lVert\bm v\rVert\quad}_\blacksquare

等号成立条件

$ \forall t\in\R\forall\bm u,\bm v\in V:

$ |\braket{\bm u|\bm v}|=\lVert\bm u\rVert\lVert\bm v\rVert

$ \implies \lVert-\lVert\bm v\rVert^2\bm u+\braket{\bm u|\bm v}\bm v\rVert^2=\lVert\bm v\rVert^4\lVert\bm u\rVert^2-2\lVert\bm v\rVert^2|\braket{\bm u|\bm v}|^2+|\braket{\bm u|\bm v}|^2\lVert\bm v\rVert^2

$ = \lVert\bm v\rVert^4\lVert\bm u\rVert^2-2\lVert\bm v\rVert^4\lVert\bm u\rVert^2+\lVert\bm v\rVert^4\lVert\bm u\rVert^2

$ =0

$ \implies-\lVert\bm v\rVert^2\bm u+\braket{\bm u|\bm v}\bm v=\bm0

$ \iff\bm v=\bm 0\lor\bm u=\frac{\braket{\bm u|\bm v}}{\lVert\bm v\rVert^2}\bm v

$ \implies\bm v=\bm 0\lor\exist\lambda\in K:\bm u=\lambda\bm v

$ \implies|\braket{\bm u|\bm v}|=0=\lVert\bm u\rVert\lVert\bm v\rVert\lor|\braket{\bm u|\bm v}|=|\braket{\lambda\bm v|\bm v}|=|\lambda||\braket{\bm v|\bm v}|=|\lambda|\lVert\bm v\rVert^2=\lVert\lambda\bm v\rVert\lVert\bm v\rVert=\lVert\bm u\rVert\lVert\bm v\rVert

$ \implies|\braket{\bm u|\bm v}|=\lVert\bm u\rVert\lVert\bm v\rVert

$ \underline{\implies\forall\bm u,\bm v\in V:|\braket{\bm u|\bm v}|=\lVert\bm u\rVert\lVert\bm v\rVert\iff\bm v=\bm 0\lor\exist\lambda\in K:\bm u=\lambda\bm v\quad}_\blacksquare

References