话题 有限维实线性空间的二次对偶空间

引入

给定 $n$ 维线性空间 $\mathscr{X} = (S,\mathbb{R})$ 及其标准正交基 $e_1,\cdots,e_n$
定义 $\mathscr{X}$ 的对偶空间为

$$ \bbox[5pt]{ \mathscr{X}^* \triangleq \mathscr{L}(\mathscr{X},\mathbb{R}) } $$

定义 $\mathscr{X}$ 的二次对偶空间为

$$ \bbox[5pt]{ \mathscr{X}^{**} \triangleq \mathscr{L}(\mathscr{X}^*,\mathbb{R}) } $$

$\mathscr{X}^*$ 上的对偶基

$\forall i \in \mathbb{N}$ 且 $1 \leqslant i \leqslant n$ ，定义 $n$ 个线性函数如下 $\displaystyle \forall x = \sum_{k=1}^{n}{x_ke_k} \in S$ $$ \bbox[5pt]{ \mathtip{\langle{f_i,x}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}} \triangleq x_i } $$

那么 $\displaystyle \forall f \in \mathtip{\mathscr{L}(\mathscr{X},\mathbb{R})}{\mathscr{X} 上的全体有界线性泛函} , \forall x = \sum_{k=1}^{n}{x_ke_k} \in S$ 成立

$$ \bbox[5pt]{ \begin{aligned} \mathtip{\langle{f,x}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}} = & \mathtip{\left\langle{f,\sum_{k=1}^{n}{x_ke_k}}\right\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}} \\ = & \sum_{k=1}^{n}{x_k\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}} \\ = & \sum_{k=1}^{n}{\mathtip{\langle{f_k,x}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}} \\ = & \sum_{k=1}^{n}{\mathtip{\langle{\mathtip{\langle{f,e_k}\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}}f_k,x}\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}}} \\ = & \mathtip{\left\langle{\sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}}f_k},x}\right\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}} \end{aligned} } $$

故

$$ \bbox[5pt]{ f = \sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}f_k} } $$

从而 $f_1,\cdots,f_n$ 构成 $\mathscr{X}^*$ 的一组基，称为 $e_1,\cdots,e_n$ 的对偶基

$\mathscr{X}^{**}$ 上的一组基

$\forall i \in \mathbb{N}$ 且 $1 \leqslant i \leqslant n$ ，定义 $n$ 个线性函数如下 $\displaystyle \forall f = \sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}f_k} \in \mathtip{\mathscr{L}(\mathscr{X},\mathbb{R})}{\mathscr{X} 上的全体有界线性泛函}$ $$ \bbox[5pt]{ \mathtip{\langle{\sigma_i,f}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}} \triangleq \mathtip{\langle{f,e_i}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}} } $$

那么 $\displaystyle \forall \sigma \in \mathtip{\mathscr{L}(\mathscr{X}^*,\mathbb{R})}{\mathscr{X}^* 上的全体有界线性泛函} , \forall f = \sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}f_k} \in \mathtip{\mathscr{L}(\mathscr{X},\mathbb{R})}{\mathscr{X} 上的全体有界线性泛函}$ 成立

$$ \bbox[5pt]{ \begin{aligned} \mathtip{\langle{\sigma,f}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}} = & \mathtip{\left\langle{\sigma,\sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\mathscr{X}^*,\mathscr{X}\to\mathbb{R}}f_k}}\right\rangle}{\mathscr{X}^{**},\mathscr{X}^*\to\mathbb{R}} \\ = & \sum_{k=1}^{n}{\mathtip{\langle{f,e_k}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}}\mathtip{\langle{\sigma,f_k}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}}} \\ = & \sum_{k=1}^{n}{\mathtip{\langle{\sigma_k,f}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}}\mathtip{\langle{\sigma,f_k}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}}} \\ = & \sum_{k=1}^{n}{\mathtip{\langle{\mathtip{\langle{\sigma,f_k}\rangle}{\mathscr{X}^{**},\mathscr{X}^*\to\mathbb{R}}\sigma_k,f}\rangle}{\mathscr{X}^{**},\mathscr{X}^*\to\mathbb{R}}} \\ = & \mathtip{\left\langle{\sum_{k=1}^{n}{\mathtip{\langle{\sigma,f_k}\rangle}{\mathscr{X}^{**},\mathscr{X}^*\to\mathbb{R}}\sigma_k},f}\right\rangle}{{\mathscr{X}^{**},\mathscr{X}^*\to\mathbb{R}}} \end{aligned} } $$

故

$$ \bbox[5pt]{ \sigma = \sum_{k=1}^{n}{\mathtip{\langle{\sigma,f_k}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}}\sigma_k} } $$

从而 $\sigma_1,\cdots,\sigma_n$ 构成 $\mathscr{X}^{**}$ 的一组基

$\mathscr{X}^{**}$ 的基与 $\mathscr{X}$ 的基之间的一一对应

回到定义

$$ \bbox[5pt]{ \mathtip{\langle{\sigma_i,f}\rangle}{\langle{\mathscr{X}^{**},\mathscr{X}^*}\rangle\to\mathbb{R}} \triangleq \mathtip{\langle{f,e_i}\rangle}{\langle{\mathscr{X}^*,\mathscr{X}}\rangle\to\mathbb{R}} } $$

我们发现可以构造 $\mathscr{X}^{**}$ 的基与 $\mathscr{X}$ 的基之间的同构映射 $\varphi : X \to \mathtip{\mathscr{L}(\mathscr{X}^*,\mathbb{R})}{\mathscr{X}^* 上的全体有界线性泛函}$ 如下

$$ \bbox[5pt]{ \varphi(e_i) = \sigma_i } $$

这样我们就可以说明

$$ \bbox[5pt]{ \mathscr{X}^{**} \cong \mathscr{X} } $$

事实上对于无穷维线性空间， $\varphi$ 虽不一定是双射，但仍然满足保留代数结构，从而是一个单同态映射