《微分流形》讨论稿-线性映照空间

张量表示

对于一般的 $r$ -重线性映照空间, 我们有

L (V_{1}, \dots, V_{r}; Z) = V_{1}^{*} \otimes \dots \otimes V_{r}^{*} \otimes Z .

$\mathcal{L}(V_1,\cdots,V_r;Z)=V_1^\ast \otimes \cdots\otimes V_r^\ast \otimes Z.$

定义 $(r+1)$ -重线性映照

\begin{matrix} f : V_{1}^{*} \times \dots \times V_{r}^{*} \times Z & \to L (V_{1}, \dots, V_{r}; Z), (v_{1}^{*}, \dots, v_{r}^{*}, z) & \mapsto v_{1}^{*} \otimes \dots \otimes v_{r}^{*} \cdot z, \end{matrix}

$\begin{aligned} f:V_1^*\times\cdots\times V_r^*\times Z&\rightarrow \mathcal{L}(V_1,\cdots,V_r;Z),\\ (v_1^*,\cdots,v_r^*,z)&\mapsto v_1^*\otimes \cdots\otimes v_r^*\cdot z, \end{aligned}$

即 $f(v_1^\ast ,\cdots,v_r^\ast ,z)(v_1,\cdots,v_r)=v_1^\ast (v_1)\cdots v_r^\ast (v_r) z=v_1^\ast \otimes \cdots \otimes v_r^\ast (v_1,\cdots,v_r) \cdot z.$ 其中 $\cdot$ 为标量乘法, 由向量空间 $Z$ 决定.

只需利用万有映照性质说明 $g:V_1^\ast \otimes \cdots\otimes V_r^\ast \otimes Z\rightarrow \mathcal{L}(V_1,\cdots,V_r;Z)$ 为同构. 首先证明

dim V_{1}^{*} \otimes \dots \otimes V_{r}^{*} \otimes Z = dim V_{1}^{*} \dots dim V_{r}^{*} dim Z = dim L (V_{1}, \dots, V_{r}; Z) .

$\dim V_1^\ast \otimes\cdots \otimes V_r^\ast \otimes Z=\dim V_1^\ast \cdots\dim V_r^\ast \dim Z=\dim\mathcal{L}(V_1,\cdots,V_r;Z).$

取 $V_1^\ast ,\cdots,V_r^\ast ,Z$ 的基 $\{\omega_{(1)}^{i_1}\},\cdots,\{\omega_{(r)}^{i_r}\},$ $\{\eta_\alpha\}.$ 那么任意 $F\in \mathcal{L}(V_1,\cdots,V_r;Z),$ 可分解为 $F^\alpha\eta_\alpha,$ $F^\alpha\in \mathcal{L}(V_1,\cdots,V_r;\mathbb{R})=V_1^\ast \otimes \cdots\otimes V_r^\ast .$ 因此每个 $F^\alpha$ 又可分解为 $a^\alpha_{i_1\cdots i_n}\omega_{(1)}^{i_1}\otimes\cdots\otimes \omega_{(n)}^{i_n}.$ 即

F = a_{i_{1} \dots i_{n}}^{α} ω_{(1)}^{i_{1}} \otimes \dots \otimes ω_{(n)}^{i_{n}} \cdot η_{α} .

$F=a^\alpha_{i_1\cdots i_n}\omega_{(1)}^{i_1}\otimes\cdots\otimes \omega_{(n)}^{i_n}\cdot\eta_\alpha.$

通过作用对偶基 $\{e^{(1)}_{i_1}\},\cdots,\{e^{(n)}_{i_n}\},$ 由 $\{\eta_\alpha\}$ 的线性无关性, 可以说明 $\{\omega_{(1)}^{i_1}\otimes\cdots\otimes \omega_{(n)}^{i_n}\cdot\eta_\alpha\}$ 的线性无关性, 从而其为一组基.

注意到,

f (ω_{(1)}^{i_{1}}, \dots, ω_{(r)}^{i_{r}}, η_{α}) = ω_{(1)}^{i_{1}} \otimes \dots \otimes ω_{(n)}^{i_{n}} \cdot η_{α} = g (ω_{(1)}^{i_{1}} \otimes \dots \otimes ω_{(r)}^{i_{r}} \otimes η_{α}),

$f(\omega^{i_1}_{(1)},\cdots,\omega^{i_r}_{(r)},\eta_\alpha)=\omega_{(1)}^{i_1}\otimes\cdots\otimes \omega_{(n)}^{i_n}\cdot\eta_\alpha=g(\omega^{i_1}_{(1)}\otimes\cdots\otimes\omega^{i_r}_{(r)}\otimes\eta_\alpha),$

$g$ 将基映到基, 从而是线性同构.

文章最后更新于 2021-11-24 22:08:11