# 内积空间

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

## 正式定义

${\displaystyle V}$ 是一個定義在 ${\displaystyle \left(F,\,+,\,\times \right)}$ 上的向量空间，其向量加法記為「 ${\displaystyle \oplus }$ 」 ，且其标量乘法記為「 ${\displaystyle \cdot }$ 」。若它裝配了一個二元函数 ${\displaystyle f:V\times V\to F}$ 滿足：（以下將 ${\displaystyle f(v,\,w)}$ 簡寫為 ${\displaystyle \langle v,\,w\rangle }$

### 定义的分歧

 线性 對所有 ${\displaystyle a,\,v,\,w\in V}$ ${\displaystyle \langle a,\,v\oplus w\rangle =\langle a,\,v\rangle +\langle a,\,w\rangle }$ 對所有 ${\displaystyle v,\,w\in V}$ 和所有 ${\displaystyle \lambda \in F}$ ${\displaystyle \langle v,\,\lambda \cdot w\rangle =\lambda \times \langle v,\,w\rangle }$

## 例子

### 点积

${\displaystyle \langle (x_{1},\ldots ,x_{n}),\,(y_{1},\ldots ,y_{n})\rangle :=\sum _{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdots +x_{n}y_{n}}$

## 基本性质

(a) ${\displaystyle \langle a,\,v\oplus w\rangle =\langle a,\,v\rangle +\langle a,\,w\rangle }$
(b) ${\displaystyle \langle v,\,\lambda \cdot w\rangle ={\overline {\lambda }}\times \langle v,\,w\rangle }$

(a)

{\displaystyle {\begin{aligned}\langle a,\,v\oplus w\rangle &={\overline {\langle v\oplus w,\,a\rangle }}\\&={\overline {\langle v,\,a\rangle +\langle w,\,a\rangle }}\\&={\overline {\langle v,\,a\rangle }}+{\overline {\langle w,\,a\rangle }}\\&=\langle a,\,v\rangle +\langle a,\,w\rangle \end{aligned}}}

(b)

{\displaystyle {\begin{aligned}\langle v,\,\lambda \cdot w\rangle &={\overline {\langle \lambda \cdot w,\,v\rangle }}\\&={\overline {\lambda \times \langle w,\,v\rangle }}\\&={\overline {\lambda }}\times {\overline {\langle w,\,v\rangle }}\\&={\overline {\lambda }}\times \langle v,\,w\rangle \end{aligned}}}

(a) ${\displaystyle \left\langle \sum _{i=1}^{n}v_{i},\,w\right\rangle =\sum _{i=1}^{n}\langle v_{i},\,w\rangle }$
(b) ${\displaystyle \left\langle w,\,\sum _{i=1}^{n}v_{i}\right\rangle =\sum _{i=1}^{n}\langle w,\,v_{i}\rangle }$

${\displaystyle n=2}$ ，本定理只是內積定义的線性部分，故成立。

${\displaystyle n=k}$ 時，對任意有限向量序列 ${\displaystyle {\{u_{i}\in V\}}_{i=1}^{k}}$ 和任意 ${\displaystyle w\in V}$ 有：

${\displaystyle \left\langle \sum _{i=1}^{k}u_{i},\,w\right\rangle =\sum _{i=1}^{k}\langle u_{i},\,w\rangle }$

{\displaystyle {\begin{aligned}\left\langle \sum _{i=1}^{k+1}v_{i},\,w\right\rangle &=\left\langle \left(\sum _{i=1}^{k}v_{i}\right)+v_{k+1},\,w\right\rangle \\&=\left\langle \sum _{i=1}^{k}v_{i},\,w\right\rangle +\langle v_{k+1},\,w\rangle \\&=\sum _{i=1}^{k}\langle v_{i},\,w\rangle +\langle v_{k+1},\,w\rangle \\&=\sum _{i=1}^{k+1}\langle v_{i},\,w\rangle \end{aligned}}}

{\displaystyle {\begin{aligned}\left\langle w,\,\sum _{i=1}^{n}v_{i}\right\rangle &={\overline {\left\langle \sum _{i=1}^{n}v_{i},\,w\right\rangle }}\\&={\overline {\sum _{i=1}^{n}\langle v_{i},\,w\rangle }}\\&=\sum _{i=1}^{n}{\overline {\langle v_{i},\,w\rangle }}\\&=\sum _{i=1}^{n}\langle w,\,v_{i}\rangle \end{aligned}}}

## 范数

${\displaystyle V}$ 是個複內積空間，則對所有的 ${\displaystyle v,\,w\in V}$ 有：

(a) ${\displaystyle \|v\|\|w\|\geq |\langle v,\,w\rangle |}$
(b) ${\displaystyle \|v\|\|w\|=|\langle v,\,w\rangle |}$ ${\displaystyle \Leftrightarrow }$ 存在 ${\displaystyle \lambda \in \mathbb {C} }$ 使 ${\displaystyle v=\lambda \cdot w}$

${\displaystyle v=w=0_{V}}$ ，根據內積定义的非退化部分，本定理成立。若考慮 ${\displaystyle v,\,w\neq 0_{V}}$ ，取 ${\displaystyle e_{v}={\frac {1}{\|v\|}}\cdot v}$${\displaystyle e_{w}={\frac {1}{\|w\|}}\cdot w}$${\displaystyle \alpha =\langle e_{v},\,e_{w}\rangle }$ ，則根據內積定义

{\displaystyle {\begin{aligned}\langle e_{v}\oplus (-\alpha )\cdot e_{w},\,e_{v}\oplus (-\alpha )\cdot e_{w}\rangle &=\langle e_{v},\,e_{v}\rangle +\langle e_{v},\,(-\alpha )\cdot e_{w}\rangle +\langle (-\alpha )\cdot e_{w},\,e_{v}\rangle +\langle (-\alpha )\cdot e_{w},\,(-\alpha )\cdot e_{w}\rangle \\&=1-{\overline {\alpha }}\alpha -\alpha {\overline {\alpha }}+\alpha {\overline {\alpha }}\\&=1-{|\alpha |}^{2}\\&=1-{\left({\frac {|\langle v,\,w\rangle |}{\|v\|\|w\|}}\right)}^{2}\\&\geq 0\end{aligned}}}

${\displaystyle e_{v}\oplus (-\alpha )\cdot e_{w}=0_{V}}$

${\displaystyle v={\frac {\langle v,\,r\rangle }{{\|w\|}^{2}}}\cdot w}$

${\displaystyle V}$ 是個複內積空間，則對所有的${\displaystyle v,\,w\in V}$ 有：

${\displaystyle \|v\|+\|w\|\geq \|v\oplus w\|}$

{\displaystyle {\begin{aligned}{\|v\oplus w\|}^{2}&=\langle v\oplus w,\,v\oplus w\rangle \\&={\|v\|}^{2}+\langle v,\,w\rangle +\langle w,\,w\rangle +{\|w\|}^{2}\\&={\|v\|}^{2}+{\|w\|}^{2}+2\operatorname {Re} (\langle v,\,w\rangle )\\&\leq {\|v\|}^{2}+{\|w\|}^{2}+2|\langle v,\,w\rangle |\end{aligned}}}

{\displaystyle {\begin{aligned}{\|v\oplus w\|}^{2}&\leq {\|v\|}^{2}+{\|w\|}^{2}+2|\langle v,\,w\rangle |\\&\leq {\|v\|}^{2}+{\|w\|}^{2}+2\|v\|\|w\|\\&={(\|v\|+\|w\|)}^{2}\end{aligned}}}

${\displaystyle \angle (v,\,w):=\cos ^{-1}\left({\frac {\langle v,\,w\rangle }{\|v\|\|w\|}}\right)}$

${\displaystyle V}$ 是個複內積空間，若有限向量序列 ${\displaystyle {\{v_{i}\in V\}}_{i=1}^{n}}$ 對任意不相等的正整數 ${\displaystyle 1\leq i\neq j\leq n}$ 都有 ${\displaystyle \langle v_{i},\,v_{j}\rangle =0}$ 則：

${\displaystyle \sum _{i=1}^{n}\|v_{i}\|^{2}=\left\|\sum _{i=1}^{n}v_{i}\right\|^{2}}$

{\displaystyle {\begin{aligned}\left\|\sum _{i=1}^{n}v_{i}\right\|^{2}&=\left\langle \sum _{i=1}^{n}v_{i},\,\sum _{i=1}^{n}v_{i}\ \right\rangle \\&=\sum _{j=1}^{n}\left\langle v_{j},\,\sum _{i=1}^{n}v_{i}\ \right\rangle \\&=\sum _{j=1}^{n}\sum _{i=1}^{n}\langle v_{j},\,v_{i}\rangle \\&=\sum _{j=1}^{n}{\|v_{j}\|}^{2}\end{aligned}}}

${\displaystyle V}$ 是個複內積空間，則對所有的 ${\displaystyle v,\,w\in V}$ 有：

(a) ${\displaystyle \|v\oplus w\|^{2}+\|v\ominus w\|^{2}=2\|v\|^{2}+2\|w\|^{2}}$

{\displaystyle {\begin{aligned}{\|v\oplus w\|}^{2}&=\langle v\oplus w,\,v\oplus w\rangle \\&={\|v\|}^{2}+\langle v,\,w\rangle +\langle w,\,w\rangle +{\|w\|}^{2}\\\end{aligned}}}

{\displaystyle {\begin{aligned}{\|v\ominus w\|}^{2}&=\langle v\oplus (w^{-1}),\,v\oplus (w^{-1})\rangle \\&={\|v\|}^{2}+\langle v,\,(-1)\cdot w\rangle +\langle (-1)\cdot w,\,v\rangle +\langle (-1)\cdot w,\,(-1)\cdot w\rangle \\&={\|v\|}^{2}-\langle v,\,w\rangle -\langle w,\,v\rangle +{\|w\|}^{2}\end{aligned}}}

## 完备化

${\displaystyle \langle f,g\rangle :=\int _{a}^{b}f(t){\overline {g(t)}}\,dt}$

${\displaystyle \forall k\geqslant 2,\;\;\;f_{k}(t)={\begin{cases}0,&\forall t\in [0,{\frac {1}{2}}]\\k(t-{\frac {1}{2}}),&\forall t\in ({\frac {1}{2}},{\frac {1}{2}}+{\frac {1}{k}}]\\1,&\forall t\in ({\frac {1}{2}}+{\frac {1}{k}},1]\end{cases}}}$

## 引用

• S. Axler, Linear Algebra Done Right, Springer, 2004
• G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley Interscience, 1972.
• N. Young, An Introduction to Hilbert Spaces, Cambridge University Press, 1988