{"id":1472,"date":"2023-05-10T00:04:53","date_gmt":"2023-05-09T16:04:53","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=1472"},"modified":"2023-05-10T00:17:02","modified_gmt":"2023-05-09T16:17:02","slug":"shixiangliangkongjiandefuhua","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=1472","title":{"rendered":"\u5b9e\u5411\u91cf\u7a7a\u95f4\u7684\u590d\u5316"},"content":{"rendered":"

1. \u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u5f20\u91cf\u79ef<\/a><\/span>\n\n1.1. \u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u4e58\u79ef\u7a7a\u95f4<\/a><\/span>\n\n\u5047\u8bbe$V,W$\u662f\u5b9e\u6570\u57df$\\mathbb{R}$\u4e0a\u7684\u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u3002\u5219\u5bb9\u6613\u9a8c\u8bc1\u4e58\u79ef\u7a7a\u95f4$V\\times W$\u4e5f\u662f\u5b9e\u5411\u91cf\u7a7a\u95f4\u3002$V\\times W$\u4e0a\u7684\u52a0\u6cd5\u4e0e\u6570\u4e58\u5b9a\u4e49\u4e3a
\n\\[
\n (v,w)+(v’,w’)=(v+v’,w+w’),\\quad
\n \\lambda(v,w)=(\\lambda v,\\lambda w).
\n\\]<\/p>\n

\u79f0\u51fd\u6570$f:V\\times W\\to \\mathbb{R}$\u662f\u53cc\u91cd\u5b9e\u7ebf\u6027\u7684\uff0c\u5982\u679c
\n\\begin{align*}
\n f( (v+v’,w) )&=f( (v,w) )+f( (v’,w) ),\\\\
\n f( (v,w+w’) )&=f( (v,w) )+f( (v,w’) ).
\n\\end{align*}
\n\u4ee5\u53ca
\n\\begin{align*}
\n f( (\\lambda v,w) )&=\\lambda f((v,w)),\\\\
\n f( (v,\\lambda w) )&=\\lambda f((v,w)).
\n\\end{align*}<\/p>\n

$V\\times W$\u4e0a\u5168\u4f53\u53cc\u91cd\u5b9e\u7ebf\u6027\u51fd\u6570\u6784\u6210\u4e00\u4e2a\u5b9e<\/span>\u5411\u91cf\u7a7a\u95f4\uff0c\u8bb0\u4f5c$(V\\times W)^*$, \u5b83\u662f$V\\times W$\u7684\u5bf9\u5076\u7a7a\u95f4<\/span>. \u4e8b\u5b9e\u4e0a\uff0c\u5b9a\u4e49\u5176\u4e0a\u7684\u52a0\u6cd5\u548c\u6570\u4e58\u5982\u4e0b
\n\\begin{align*}
\n (f+g) \\left((v,w)\\right)&=f\\left( (v,w) \\right)+ g\\left( (v,w)\\right),\\\\
\n (\\lambda f)\\left( (v,w) \\right)&=f\\left( \\lambda(v,w) \\right)=\\lambda f\\left( (v,w) \\right).
\n\\end{align*}
\n\u5bb9\u6613\u9a8c\u8bc1\uff0c$(V\\times W)^*$\u5728\u4e0a\u8ff0\u52a0\u6cd5\u548c\u6570\u4e58\u4e0b\u6210\u4e3a\u4e00\u4e2a\u5b9e\u5411\u91cf\u7a7a\u95f4\u3002
\n
\n1.2. \u5173\u4e8e\u5bf9\u5076\u7a7a\u95f4\u7684\u57fa\u672c\u4e8b\u5b9e<\/a><\/span>\n\n\u5047\u8bbe$V$\u662f\u5b9e\u5411\u91cf\u7a7a\u95f4\uff0c\u5b83\u6709\u57fa\u5e95$V=\\mathrm{span}\\left\\{ e_i \\right\\}_{i=1}^n$, \u5219\u5176\u5bf9\u5076\u7a7a\u95f4\uff08\u5373$V$\u4e0a\u5168\u4f53\u7ebf\u6027\u6620\u5c04\u6784\u6210\u7684\u5b9e\u5411\u91cf\u7a7a\u95f4)$V^*$\u6709\u57fa\u5e95$V^*=\\mathrm{span}\\left\\{ e_i^* \\right\\}_{i=1}^n$. \u8fd9\u91cc\uff0c$e_i^*\\in V^*$\u5b9a\u4e49\u4e3a
\n\\[
\n e_i^*(e_j)=\\delta_{ij},
\n\\]
\n\u6545, \u82e5$v=\\sum_{i=1}^n\\lambda_ie_i$, \u5219
\n\\[
\n e_i^*(v)=e_i^*\\left( \\sum_{j=1}^n\\lambda_je_j \\right)=\\sum_{j=1}^n\\lambda_j e_i^*(e_j)=\\sum_{j=1}^n\\lambda_j\\delta_{ij}=\\lambda_i.
\n\\]
\n\u56e0\u6b64\uff0c $e_i^*$\u4e5f\u79f0\u4e3a\u53d6\u5750\u6807\u6620\u5c04<\/span>\u3002
\n

\u547d\u9898 1<\/span>.<\/span> \u5047\u8bbe$V$\u662f\u5b9e\u5411\u91cf\u7a7a\u95f4\uff0c\u5219\u53d6\u5750\u6807\u6620\u5c04\u8bf1\u5bfc\u4e86\u4e00\u4e2a$V\\to V^*$\u7684\u7ebf\u6027\u540c\u6784\u6620\u5c04\u3002
\n<\/div>
\n\u4e8b\u5b9e\u4e0a\uff0c\u5b9a\u4e49$\\tau: V\\to V^*$, $v=\\lambda_ie_i\\mapsto \\lambda_ie_i^*=\\tau(v)=:v^*$. \u5219\u5bb9\u6613\u6839\u636e$V^*$\u4e2d\u52a0\u6cd5\u548c\u6570\u4e58\u7684\u5b9a\u4e49\u9a8c\u8bc1$\\tau(v+v’)=\\tau(v)+\\tau(v’)$\u4ee5\u53ca$\\tau(\\lambda v)=\\lambda\\tau(v)$. <\/p>\n

\u6b64\u5916\uff0c$\\tau$\u8fd8\u4e0d\u4f9d\u8d56\u4e8e\u57fa\u5e95\u7684\u9009\u53d6\uff0c\u5373\u82e5$v=\\sum_i\\lambda_ie_i=\\sum_j\\tilde{\\lambda}_j\\tilde e_j$, \u5219$\\tau(v)=\\sum_i\\lambda_ie_i^*=\\sum_j\\tilde{\\lambda}_j \\tilde{e_j}^*$. \u9a8c\u8bc1\u5982\u4e0b:
\n\\begin{align*}
\n \\tilde{e}_j&=\\sum_ie_ia_{ij},\\quad v=\\sum_i\\lambda_ie_i= \\sum_j \\tilde{\\lambda}_j \\tilde{e}_j=\\sum_j \\tilde{\\lambda}_j\\sum_ia_{ji}e_i,\\\\
\n \\lambda_i&=\\sum_j \\tilde{\\lambda}_ja_{ji}\\iff \\tilde{\\lambda}_j=\\sum_ib_{ji}\\lambda_i, B=(b_{ij})=A^{-1}, A=(a_{ij})\\\\
\n \\sum_j\\tilde{\\lambda}_j \\tilde{e}_j^*
\n\t &=\\sum_j\\left( \\sum_ib_{ji}\\lambda_i \\right)\\left( \\sum_ka_{kj}e_k \\right)^*
\n\t =\\sum_j\\left( \\sum_ib_{ji}\\lambda_i \\right)\\tau\\left( \\sum_ka_{kj}e_k \\right)\\\\
\n\t &=\\sum_j\\left( \\sum_ib_{ji}\\lambda_i \\right) \\sum_ka_{kj}\\tau\\left(e_k \\right)
\n\t =\\sum_j\\left( \\sum_ib_{ji}\\lambda_i \\right) \\sum_ka_{kj}e_k^*\\\\
\n\t &=\\sum_{i,j,k}a_{kj}b_{ji}\\lambda_ie_k^*=\\sum_{i,j,k}\\delta_{ki}\\lambda_ie_k^*=\\sum_i\\lambda_ie_i^*.
\n\\end{align*}
\n\u56e0\u6b64\u6211\u4eec\u79f0$\\tau$\u4e3a\u4e00\u4e2a\u5178\u5219\u540c\u6784<\/span>\u3002<\/p>\n

\u4e0b\u9762\uff0c\u6211\u4eec\u6765\u8bf4\u660e$V^*$\u7684\u5bf9\u5076\u7a7a\u95f4\u53ef\u4ee5\u81ea\u7136\u7b49\u540c\u4e8e$V$. \u4e8b\u5b9e\u4e0a\uff0c \u5047\u8bbe$\\varphi\\in (V^*)^*$, \u5373$V^*$\u4e0a\u7684\u5b9e\u7ebf\u6027\u51fd\u6570\uff0c\u5219\u5bf9\u4efb\u610f\u7684$v^*=\\sum_{i=1}^n\\lambda_ie_i^*\\in V^*$, \u6211\u4eec\u6709
\n\\begin{align*}
\n \\varphi(v^*)&=\\sum_{i=1}^n\\lambda_i\\varphi(e_i^*)=\\sum_{i=1}^n\\varphi(e_i^*)\\sum_{j=1}^n\\lambda_j\\delta_{ij}
\n =\\sum_{i=1}^n\\varphi(e_i^*)\\sum_{j=1}^n\\lambda_je_j^*(e_i)\\\\
\n\t &=\\sum_{i=1}^n\\varphi(e_i^*)v^*(e_i)=v^*\\left( \\sum_{i=1}^n\\varphi(e_i^*)e_i \\right),
\n\\end{align*}
\n\u56e0\u6b64\uff0c\u82e5\u5b9a\u4e49$v_\\varphi=\\sum_{i=1}^n\\varphi(e_i^*)e_i$, \u5219\u5bf9\u4efb\u610f\u7684$v^*\\in V^*$, \u6211\u4eec\u90fd\u6709
\n\\[
\n \\varphi(v^*)=v^*(v_\\varphi).
\n\\]
\n\u73b0\u5728\uff0c\u5982\u679c\u6211\u4eec\u5b9a\u4e49
\n\\[
\n \\left\\langle v_\\varphi,v^* \\right\\rangle=v^*(v_\\varphi)=\\varphi(v^*),
\n\\]
\n\u5219$ \\left\\langle v_\\varphi,\\cdot \\right\\rangle$\u4e8b\u5b9e\u4e0a\u8bf1\u5bfc\u4e86$V^*$\u4e0a\u4e00\u4e2a\u7ebf\u6027\u51fd\u6570: \u76f4\u63a5\u9a8c\u8bc1\u77e5\u9053
\n\\begin{align*}
\n \\left\\langle v_\\varphi,v_1^*+v_2^* \\right\\rangle&=(v_1^*+v_2^*)(v_\\varphi)=v_1^*(v_\\varphi)+v_2^*(v_\\varphi)= \\left\\langle v_\\varphi,v_1^* \\right\\rangle+ \\left\\langle v_\\varphi,v_2^* \\right\\rangle,\\\\
\n \\left\\langle v_\\varphi,\\lambda v^* \\right\\rangle&=(\\lambda v^*)(v_\\varphi)=v^*(\\lambda v_\\varphi)=\\lambda v^*(v_\\varphi)=\\lambda \\left\\langle v_\\varphi,v^* \\right\\rangle.
\n\\end{align*}
\n\u56e0\u6b64\uff0c$\\varphi= \\left\\langle v_\\varphi,\\cdot \\right\\rangle$. \u7279\u522b\u5730\uff0c\u6211\u4eec\u5f97\u5230\u540c\u6784\u6620\u5c04
\n\\[
\n \\sigma: (V^*)^*\\to V,\\quad \\varphi\\mapsto v_\\varphi.
\n\\]<\/p>\n

\u5173\u4e8e\u4e0a\u8ff0\u6620\u5c04$\\sigma$\u7684\u7ebf\u6027\u6027\u9a8c\u8bc1\u5982\u4e0b\uff1a\u5bf9\u4efb\u610f\u7684$\\varphi_1,\\varphi_2\\in (V^*)^*$, \u4ee5\u53ca\u4efb\u610f\u7684\u5b9e\u6570$\\lambda\\in \\mathbb{R}$, \u6709
\n\\begin{align*}
\n v_{\\varphi_1+\\lambda\\varphi_2}&=\\sum_i(\\varphi_1+\\lambda\\varphi_2)(e_i^*)e_i
\n =\\sum_i\\left( \\varphi_1(e_i^*)+\\lambda\\varphi_2(e_i^*) \\right)e_i\\\\
\n\t\t\t\t&=\\sum_i\\left( \\varphi_1(e_i^*)e_i+\\lambda\\varphi_2(e_i^*)e_i \\right)
\n\t\t\t\t=v_{\\varphi_1}+\\lambda v_{\\varphi_2}.
\n\\end{align*}
\n\u5219\u8868\u660e
\n\\[
\n \\sigma(\\varphi_1+\\lambda \\varphi_2)=v_{\\varphi_1+\\lambda \\varphi_2}=v_{\\varphi_1}+\\lambda v_{\\varphi_2}=\\sigma(\\varphi_1)+\\lambda \\sigma(\\varphi_2).
\n\\]
\n\u5373$\\sigma$\u662f\u7ebf\u6027\u6620\u5c04\u3002<\/p>\n

\u5982\u679c\u6211\u4eec\u5b9a\u4e49$V^*$\u4e0a\u7684\u53d6\u5750\u6807\u6620\u5c04$(e_i^*)^*(e_j^*)=\\delta_{ij}$, \u5219
\n\\[
\n v_{(e_i^*)^*}=\\sum_j(e_i^*)^*(e_j^*)e_j=\\sum_j\\delta_{ij}e_j=e_i.
\n\\]
\n\u5219\u8868\u660e$\\sigma((e_i^*)^*)=e_i$, \u4ece\u800c$\\sigma\\left(\\sum_i\\lambda_i(e_i^*)^*\\right)=\\sum_i\\lambda_i\\sigma\\left((e_i^*)^*\\right)=\\sum_i\\lambda_ie_i$. \u53ef\u89c1\u82e5\u4ee4\u53d6\u5750\u6807\u6620\u5c04\u8bf1\u5bfc\u7684\u540c\u6784\u5206\u522b\u4e3a$\\tau_1: V\\to V^*$, $\\tau_2:V^*\\to (V^*)^*$, \u5219$\\sigma^{-1}=\\tau_2\\tau_1$, \u5b83\u662f\u4e0e\u57fa\u5e95\u9009\u53d6\u65e0\u5173\u7684\u81ea\u7136\u540c\u6784.<\/p>\n

\u603b\u7ed3\u8d77\u6765\uff0c\u6211\u4eec\u5f97\u5230
\n

\u547d\u9898 2<\/span>.<\/span> \u5047\u8bbe$V$\u662f\u5411\u91cf\u7a7a\u95f4\u3002\u5219\uff0c\u53d6\u5750\u6807\u6620\u5c04\u53ef\u4ee5\u8bf1\u5bfc$V\\to (V^*)^*$\u4e4b\u95f4\u7684\u540c\u6784\u6620\u5c04\u3002
\n<\/div>
\n1.3. \u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u5f20\u91cf\u79ef<\/a><\/span>\n\n\u73b0\u5728\u5047\u8bbe$V,W$\u662f\u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\uff0c\u5176\u5bf9\u5076\u7a7a\u95f4\u5206\u522b\u4e3a$V^*,W^*$. \u7ed9\u5b9a$v\\in V, w\\in W$, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49$V^*\\times W^*$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570$v\\otimes w$, \u5982\u4e0b
\n\\[
\n (v\\otimes w)(v^*,w^*):=\\sigma^{-1}(v)(v^*)\\cdot\\sigma^{-1}(w)(w^*)= \\left\\langle v,v^* \\right\\rangle\\cdot \\left\\langle w,w^* \\right\\rangle.
\n\\]
\n\u6309\u7167$ \\left\\langle \\cdot,\\cdot \\right\\rangle$\u7684\u53cc\u7ebf\u6027\u5bb9\u6613\u77e5\u9053$v\\otimes w$\u786e\u5b9e\u662f$V^*\\times W^*$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570\u3002\u6211\u4eec\u5c06$V^*\\times W^*$\u4e0a\u7684\u5b9e\u53cc\u7ebf\u6027\u51fd\u6570\u5168\u4f53\u8bb0\u4e3a$\\mathscr{L}(V^*,W^*)$. \u6ce8\u610f$V^*\\times W^*$\u4e0a\u7684\u5b9e\u53cc\u7ebf\u6027\u51fd\u6570\u4e0d\u4e00\u5b9a\u662f$V^*\\times W^*$\u4e0a\u7684\u7ebf\u6027\u51fd\u6570, \u53cd\u4e4b\u4ea6\u7136\u3002\u6545$\\mathscr{L}(V^*,W^*)$\u4e0e\u5bf9\u5076\u7a7a\u95f4$(V^*\\times W^*)^*$\u6ca1\u6709\u76f8\u4e92\u5305\u542b\u5173\u7cfb\u3002<\/p>\n

\u8fdb\u4e00\u6b65\u5730\uff0c\u540c\u6837\u6839\u636e$ \\left\\langle \\cdot,\\cdot \\right\\rangle$\u7684\u53cc\u7ebf\u6027\u5bb9\u6613\u77e5\u9053,
\n

\u547d\u9898 3<\/span>.<\/span> \u5047\u8bbe$V,W$\u662f\u5411\u91cf\u7a7a\u95f4\u3002 \u5219\uff0c
\n \\[
\n \\otimes: V\\times W\\to \\mathscr{L}(V^*, W^*)
\n\\]
\n\u662f$V\\times W$\u4e0a\u7684\u53cc\u7ebf\u6027\u6620\u5c04.
\n<\/div>
\n\u7531\u6b64\uff0c\u6211\u4eec\u5f97\u5230$V$\u4e0e$W$\u7684\u5f20\u91cf\u7a7a\u95f4\u7684\u5b9a\u4e49
\n
\u5b9a\u4e49 4<\/span>.<\/span> \u5047\u8bbe$V,W$\u662f\u5411\u91cf\u7a7a\u95f4\u3002\u6211\u4eec\u79f0
\n \\[
\n V\\otimes W=\\mathrm{span}\\left\\{ v\\otimes w: v\\in V,w\\in W \\right\\},
\n \\]
\n \u4e3a$V\\times W$\u7684\u5f20\u91cf\u7a7a\u95f4<\/span>, \u5b83\u662f$\\mathscr{L}(V^*, W^*)$\u7684\u5b50\u7a7a\u95f4\u3002
\n<\/div>
\n\u5047\u8bbe$V=\\mathrm{span}\\left\\{ e_i \\right\\}_{i=1}^n$, $W=\\mathrm{span}\\left\\{ f_\\alpha \\right\\}_{\\alpha=1}^m$, \u5176\u4e2d$\\left\\{ e_i \\right\\}_{i=1}^n$, $\\left\\{ f_\\alpha \\right\\}_{\\alpha=1}^m$\u5206\u522b\u662f\u57fa\u5e95, \u76f8\u5e94\u5bf9\u5bf9\u5076\u57fa\u7528*\u8868\u793a\u3002 \u5219\u5bf9$V\\ni v=\\sum_i \\lambda_ie_i=\\sum_i e_i^*(v)e_i$, $W\\ni w=\\sum \\mu_\\alpha f_\\alpha=f_\\alpha^*(w)f_\\alpha$, \u5229\u7528$\\otimes$\u7684\u53cc\u7ebf\u6027\u6027\uff0c\u6211\u4eec\u6709
\n\\[
\n v\\otimes w=\\sum_{i,\\alpha}\\lambda_i\\mu_\\alpha e_i\\otimes f_\\alpha
\n =\\sum_{i,\\alpha}e_i^*(v)f_\\alpha^*(w)e_i\\otimes f_\\alpha.
\n\\]
\n\u56e0\u6b64\uff0c$V\\otimes W$\u4e5f\u53ef\u4ee5\u7531$\\left\\{ e_i\\otimes f_\\alpha \\right\\}_{i,\\alpha}$\u751f\u6210\u3002\u6b64\u5916\uff0c\u8fd9\u4e2a\u751f\u6210\u5143\u96c6\u5408\u8fd8\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002\u4e8b\u5b9e\u4e0a\uff0c\u82e5
\n\\[
\n \\sum_{i,\\alpha}a_{i,\\alpha}e_i\\otimes f_\\alpha=0,
\n\\]
\n\u5219
\n\\[
\n 0=\\left( \\sum_{i,\\alpha}a_{i,\\alpha}e_i\\otimes f_\\alpha \\right)(e_j^*,f_\\beta^*)=\\sum_{i,\\alpha}a_{i,\\alpha}\\left( e_i\\otimes f_\\alpha(e_j^*,f_\\beta^*) \\right)=\\sum_{i,\\alpha}a_{i,\\alpha}\\delta_{ij}\\delta_{\\alpha\\beta}=a_{j,\\beta}.
\n\\]
\n\u56e0\u6b64\uff0c\u6211\u4eec\u8bc1\u660e\u4e86$V\\otimes W$\u7684\u4e00\u4e2a\u57fa\u5e95\u4e3a$\\left\\{ e_i\\otimes f_\\alpha \\right\\}_{i,\\alpha}$. <\/p>\n

\u7531\u4e8e$\\dim \\mathscr{L}(V^*, W^* )=\\dim V^*\\times W^*=\\dim V\\times\\dim W$, \u6211\u4eec\u77e5\u9053
\n\\[
\n V\\otimes W=\\mathscr{L}(V^*,W^*).
\n\\]<\/p>\n

\u6700\u540e\uff0c\u6211\u4eec\u6765\u8ba8\u8bba$V\\otimes W$\u7684\u5bf9\u5076\u7a7a\u95f4\u3002\u6ce8\u610f\u5230\uff0c\u4e0a\u9762\u5173\u4e8e\u5f20\u91cf\u7a7a\u95f4$V\\otimes W$\u7684\u5b9a\u4e49, \u5b8c\u5168\u53ef\u4ee5\u7c7b\u4f3c\u5b9a\u4e49\u5f20\u91cf\u7a7a\u95f4$V^*\\otimes W^*$. \u7279\u522b\u5730\u6309\u7167$V^*\\otimes W^*$\u7684\u5b9a\u4e49\uff0c\u5bf9\u4efb\u610f\u7684$v^*\\in V^*, w^*\\in W^*$, \u6211\u4eec\u77e5\u9053$v^*\\otimes w^*\\in V^*\\otimes W^*$, \u800c\u4e14\u5b83\u662f$(V^*)^*\\times (W^*)^*$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570, \u5176\u5b9a\u4e49\u5982\u4e0b, \u5bf9$v\\in V, w\\in W$,
\n\\[
\n (v^*\\otimes w^*)(\\sigma^{-1}(v),\\sigma^{-1}(w)):= \\sigma^{-1}(v)(v^*) \\cdot \\sigma^{-1}(w)(w^*)
\n = \\left\\langle v,v^* \\right\\rangle\\cdot \\left\\langle w,w^* \\right\\rangle.
\n\\]
\n\u7531\u6b64\uff0c\u6211\u4eec\u5176\u5b9e\u53ef\u4ee5\u5b9a\u4e49
\n\\[
\n (v^*\\otimes w^*)(v,w):= \\left\\langle v,v^* \\right\\rangle\\cdot \\left\\langle w,w^* \\right\\rangle,
\n\\]
\n\u5b83\u662f$V\\times W$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570\u3002\u95ee\u9898\u662f\uff1a\u6211\u4eec\u662f\u5426\u53ef\u4ee5\u5229\u7528\u8fd9\u4e2a\u53cc\u7ebf\u6027\u51fd\u6570\u5b9a\u4e49\u4e00\u4e2a\u5728$V\\otimes W$\u4e0a\u7684\u7ebf\u6027\u51fd\u6570<\/span>\uff0c\u4f7f\u5f97\u4e0b\u9762\u7684\u56fe\u4ea4\u6362
\n\\begin{CD}
\nV\\times W@>v^*\\otimes w^*>> \\mathbb{R}\\\\
\n@VV\\otimes V @AA {\\exists!g} A\\\\
\nV\\otimes W @>\\mathrm{id}>> V\\otimes W
\n\\end{CD}
\n\u8fd9\u6b63\u662f\u4e0b\u9762\u5b9a\u7406\u8bc1\u660e\u7684\uff0c\u5f20\u91cf\u7a7a\u95f4\u5177\u6709\u6cdb\u6027\u8d28<\/span>(Universal property). \u5373
\n

\u5b9a\u7406 5<\/span>.<\/span> \u5047\u8bbe$V,W,Z$\u662f\u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u3002\u5982\u679c\u4ee4$\\varphi: V\\times W\\to V\\otimes W$\u662f\u7531\u5f20\u91cf\u79ef$\\otimes$\u7ed9\u51fa\u7684\u53cc\u7ebf\u6027\u6620\u5c04\uff0c\u5373
\n \\[
\n \\varphi(v,w)=v\\otimes w,\\quad v\\in V,w\\in W.
\n \\]
\n \u5219\uff0c\u5bf9\u4efb\u610f\u7684\u53cc\u7ebf\u6027\u6620\u5c04$h: V\\times W\\to Z$, \u90fd\u5b58\u5728\u552f\u4e00\u7684\u7ebf\u6027\u6620\u5c04$ \\tilde{h}: V\\otimes W\\to Z$\u4f7f\u5f97$h= \\tilde{h}\\circ \\varphi$. \u5373\u4e0b\u56fe\u4ea4\u6362\uff1a
\n\"universal_tensor_prod.svg\"<\/a><\/p>\n

\u6211\u4eec\u79f0\u5f20\u91cf\u79ef$\\otimes$\u5177\u6709\u6cdb\u6027\u8d28\u3002
\n<\/div>
\n

\u8bc1\u660e<\/span>.<\/span> \u5b9a\u4e49\u7ebf\u6027\u6620\u5c04$ \\tilde{h}: V\\otimes W\\to Z$, \u4f7f\u5f97\u5b83\u5728\u57fa\u5e95\u4e0a\u7684\u4f5c\u7528\u662f
\n \\[
\n \\tilde{h}(e_i\\otimes f_\\alpha)=h(e_i,f_\\alpha).
\n \\]
\n \u5219\u5bb9\u6613\u9a8c\u8bc1, \u5bf9$v=\\sum_i \\lambda_ie_i$, $w=\\sum_\\alpha \\mu_\\alpha f_\\alpha$, \u6709(\u6839\u636e$\\otimes$\u7684\u53cc\u7ebf\u6027)$v\\otimes w=\\sum_{i,\\alpha}\\lambda_i\\mu_\\alpha e_i\\otimes f_\\alpha$, \u4ece\u800c
\n \\[
\n \\tilde{h}(v\\otimes w)=\\sum_{i,\\alpha}\\lambda_i\\mu_\\alpha \\tilde{h}(e_i\\otimes f_\\alpha)
\n =\\sum_{i,\\alpha}\\lambda_i\\mu_\\alpha h(e_i,f_\\alpha)
\n =h(v,w).
\n \\]
\n \u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u6211\u4eec\u5229\u7528\u4e86$h$\u7684\u53cc\u7ebf\u6027\u6027\u3002 \u56e0\u6b64\uff0c$h= \\tilde{h}\\circ \\varphi$. \u660e\u663e\u5730\uff0c\u8fd9\u6837\u6784\u9020\u7684$ \\tilde{h}$\u662f\u552f\u4e00\u7684\u3002
\n<\/div>
\n\u5229\u7528\u5f20\u91cf\u79ef\u7684\u6cdb\u6027\u8d28\uff0c\u6211\u4eec\u77e5\u9053$v^*\\times w^*$\u4f5c\u4e3a$V\\times W$\u4e0a\u7684\u53cc\u7ebf\u6027\u51fd\u6570\u53ef\u4ee5\u552f\u4e00\u8bf1\u5bfc$V\\otimes W$\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u51fd\u6570<\/span>, \u5373$(V\\otimes W)^*$\u4e2d\u4e00\u4e2a\u5143\u7d20\u3002\u8fd9\u6837\uff0c\u6211\u4eec\u5c06$V^*\\otimes W^*$\u5d4c\u5165\u5230$(V\\otimes W)^*$. \u6839\u636e\u7ef4\u6570\u516c\u5f0f$\\dim (V^*\\otimes W^*)=\\dim V^*\\cdot \\dim W^*=\\dim V\\cdot \\dim W=\\dim (V\\otimes W)^*$, \u6211\u4eec\u77e5\u9053$V^*\\otimes W^*\\cong (V\\otimes W)^*$.
\n2. \u5b9e\u5411\u91cf\u7a7a\u95f4\u7684\u590d\u5316<\/a><\/span>\n\n\u73b0\u5728\u8003\u5bdf\u7279\u6b8a\u7684$W$, \u5b83\u5c31\u662f\u590d\u6570\u57df$\\mathbb{C}$, \u5f53\u7136\u53ef\u4ee5\u89c6\u4e3a2\u7ef4\u5b9e\u5411\u91cf\u7a7a\u95f4\u3002\u56e0\u6b64\u6709\u5f20\u91cf\u7a7a\u95f4\uff08\u5b9e\u5411\u91cf\u7a7a\u95f4\uff09$V\\otimes_{\\mathbb{R}}\\mathbb{C}$, \u8fd9\u91cc\u53ea\u662f\u4e3a\u4e86\u5f3a\u8c03\u8be5\u5f20\u91cf\u7a7a\u95f4\u7684\u6570\u4e58\u662f\u5b9e\u6570\u3002\u4e00\u4ef6\u6709\u610f\u601d\u7684\u4e8b\u60c5\u662f\u6211\u4eec\u53ef\u4ee5\u6070\u5f53\u7684\u5b9a\u4e49\u590d\u6570\u4e58\u6cd5\uff0c\u4f7f\u5f97\u5b83\u6210\u4e3a\u4e00\u4e2a\u590d\u5411\u91cf\u7a7a\u95f4<\/span>\u3002<\/p>\n

\u4e3a\u6b64\uff0c\u5bf9\u4efb\u610f\u7ed9\u5b9a\u7684$z_0\\in \\mathbb{C}$, \u8003\u5bdf\u6620\u5c04$\\rho_{z_0}: V\\times \\mathbb{C}\\to V\\otimes_{\\mathbb{R}}\\mathbb{C}$, $(v,z)\\mapsto v\\otimes(zz_0)$.
\n

\u65ad\u8a00 1<\/span>.<\/span> $\\rho_{z_0}$\u662f\u5b9e\u53cc\u91cd\u7ebf\u6027\u7684\u3002
\n<\/div>
\n
\u8bc1\u660e<\/span>.<\/span> \u4e8b\u5b9e\u4e0a\uff0c\u53ea\u9700\u9a8c\u8bc1
\n \\begin{align*}
\n \\rho_{z_0}( (v+v’,z) )&=(v+v’)\\otimes (zz_0)\\\\
\n\t\t\t &=v\\otimes (zz_0)+v’\\otimes (zz_0)\\\\
\n\t\t\t &=\\rho_{z_0}( (v,z))+\\rho_{z_0}( (v’,z)),
\n \\end{align*}
\n \u8fd9\u91cc\u7b2c\u4e8c\u4e2a\u7b49\u53f7\u7528\u5230\u4e86$\\otimes$\u662f\u53cc\u91cd\u7ebf\u6027\u6620\u5c04\u3002 \u6b64\u5916, \u540c\u6837\u5229\u7528\u5f20\u91cf\u79ef\u5173\u4e8e\u7b2c\u4e8c\u4e2a\u5206\u91cf\u7684\u7ebf\u6027\u6027\uff0c\u6709
\n \\begin{align*}
\n \\rho_{z_0}( (v, z+z’) )&=v \\otimes (z+z’)z_0=v\\otimes zz_0+v\\otimes z’z_0=\\rho_{z_0}(v,z)+\\rho_{z_0}(v,z’).
\n \\end{align*}
\n \u56e0\u6b64\uff0c$\\rho_{z_0}$\u662f\u53cc\u91cd\u7ebf\u6027\u6620\u5c04\u3002
\n<\/div>
\n\u5229\u7528\u5f20\u91cf\u79ef\u7684\u6cdb\u6027\u8d28\uff0c\u6211\u4eec\u77e5\u9053\u5b58\u5728\u552f\u4e00\u7684\u7ebf\u6027\u6620\u5c04$ \\tilde{\\rho}_{z_0}: V \\otimes_{\\mathbb{R}} \\mathbb{C}\\to V \\otimes_{\\mathbb{R}} \\mathbb{C}$, \u4f7f\u5f97$\\rho_{z_0}= \\tilde{\\rho}_{z_0} \\circ \\varphi$, \u5373
\n\\[
\n \\tilde{\\rho}_{z_0}(v\\otimes z)=\\rho_{z_0}(v,z)=v\\otimes (zz_0).
\n\\]
\n\u56e0\u6b64\uff0c\u4f46$z_0\\in \\mathbb{C}$\u53d6\u904d\u65f6\uff0c\u6211\u4eec\u5f97\u5230\u6620\u5c04
\n\\[
\n (V\\otimes_{\\mathbb{R}}\\mathbb{C})\\times \\mathbb{C}\\to V\\otimes_{\\mathbb{R}}\\mathbb{C},\\quad
\n (v\\otimes z,z’)\\mapsto v\\otimes (zz’).
\n\\]
\n\u8fd9\u5c31\u5b9a\u4e49\u4e86$V \\otimes_{\\mathbb{R}}\\mathbb{C}$\u4e0a\u7684$\\mathbb{C}$-\u6570\u4e58\u3002\u5373
\n\\[
\n z’\\cdot (v\\otimes z)=v\\otimes (zz’).
\n\\]
\n\u7ed3\u5408$V\\otimes_{\\mathbb{R}}\\mathbb{C}$\u539f\u6765\u7684\u52a0\u6cd5\uff0c$V \\otimes_{\\mathbb{R}}\\mathbb{C}$\u6210\u4e3a\u4e00\u4e2a\u590d\u5411\u91cf\u7a7a\u95f4\uff0c\u79f0\u4e3a\u5b9e\u5411\u91cf\u7a7a\u95f4$V$\u7684\u590d\u5316, \u5c06\u5176\u7b80\u8bb0\u4e3a$V_{\\mathbb{C}}$\u3002<\/p>\n

\u5bb9\u6613\u77e5\u9053\u590d\u5411\u91cf\u7a7a\u95f4$V_{\\mathbb{C}}$\u4f5c\u4e3a\u590d\u6570\u57df\u4e0a\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u6709\u57fa\u5e95$\\left\\{ e_i\\otimes 1 \\right\\}$. \u4e8b\u5b9e\u4e0a\uff0c \u4f5c\u4e3a\u96c6\u5408$V_{\\mathbb{C}}$\u548c$V\\otimes_{\\mathbb{R}}\\mathbb{C}$\u76f8\u540c\uff0c\u5373\u5b83\u662f\u6240\u6709\u5f62\u5982$\\left\\{ v\\otimes z:v\\in V, z\\in \\mathbb{C} \\right\\}$ \u7684\u5b9e\u7ebf\u6027\u7ec4\u5408\u5f62\u6210\u7684\u96c6\u5408\u3002\u6ce8\u610f\u5230\uff0c$\\mathbb{C}$\u4f5c\u4e3a\u5b9e\u5411\u91cf\u7a7a\u95f4\u6709\u57fa\u5e95$\\left\\{ 1, \\sqrt{-1} \\right\\}$, \u56e0\u6b64\u4efb\u610f$V_{\\mathbb{C}}$\u4e2d\u7684\u5143\u7d20\u53ef\u8868\u793a\u4e3a
\n\\[
\n \\sum_i\\left(\\lambda_i(e_i\\otimes 1)+\\mu_i(e_i\\otimes \\sqrt{-1})\\right),
\n\\]
\n\u5176\u4e2d$\\lambda_i,\\mu_i\\in \\mathbb{R}$. \u73b0\u5728\u6ce8\u610f\u5230\uff0c\u6309\u7167$V_{\\mathbb{C}}$\u4e2d\u590d-\u6570\u4e58\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u6709
\n\\begin{align*}
\n \\sum_i (\\lambda_i+ \\sqrt{-1}\\mu_i)\\cdot (e_i\\otimes 1)&=\\sum_ie_i\\otimes (1\\cdot \\left( \\lambda_i+ \\sqrt{-1}\\mu_i \\right)\\\\
\n\t\t\t\t\t\t\t&=\\sum_{i}e_i \\otimes \\left( \\lambda_i+ \\sqrt{-1}\\mu_i \\right)\\\\
\n\t\t\t\t\t\t\t&=\\sum_i \\left(\\lambda_i (e_i \\otimes 1)+ \\mu_i e_i \\otimes \\sqrt{-1}\\right).
\n\\end{align*}
\n\u6700\u540e\u4e00\u6b65\u5229\u7528\u4e86\u5f20\u91cf\u79ef\u7684\u591a\u91cd\u7ebf\u6027\u6027\u3002\u53ef\u89c1$V_{\\mathbb{C}}=\\mathrm{span}\\left\\{ e_i\\otimes 1: e_i\\in V \\right\\}$. \u6700\u540e\uff0c\u8fd8\u53ef\u4ee5\u9a8c\u8bc1\u8be5\u751f\u6210\u96c6\u5408\u662f\u7ebf\u6027\u65e0\u5173\u7684\uff0c\u4ece\u800c$V_{\\mathbb{C}}$\u4f5c\u4e3a\u590d\u5411\u91cf\u7a7a\u95f4\uff0c\u5b83\u7684\u7ef4\u6570\u548c$V$\u7684\u7ef4\u6570\u76f8\u540c\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5047\u8bbe\u5b58\u5728\u590d\u6570$z_i=a_i+ \\sqrt{-1}b_i$, $a_i,b_i\\in \\mathbb{R}$, \u4f7f\u5f97
\n\\[
\n 0=\\sum_iz_i\\cdot (e_i\\otimes 1)=\\sum_i e_i\\otimes 1\\cdot z_i=\\sum_{i}e_i \\otimes (a_i+ \\sqrt{-1}b_i)
\n =\\sum_i \\left( a_i(e_i \\otimes 1)+b_i (e_i \\otimes \\sqrt{-1}) \\right),
\n\\]
\n\u7531\u4e8e$V_{\\mathbb{C}}=V\\otimes_{\\mathbb{R}}\\mathbb{C}$\u4f5c\u4e3a\u5b9e\u5411\u91cf\u7a7a\u95f4\uff0c$\\left\\{ e_i\\otimes 1,e_i\\otimes \\sqrt{-1} \\right\\}$\u4e3a\u5176\u57fa\u5e95\uff0c\u53ef\u89c1$a_i=0=b_i$\u5bf9\u6240\u6709\u7684$i$\u90fd\u6210\u7acb\u3002\u6545$z_i\\equiv0$. \u8fd9\u5c31\u9a8c\u8bc1\u4e86$\\left\\{ e_i \\otimes1 \\right\\}$\u662f\u590d\u5411\u91cf\u7a7a\u95f4$V_{\\mathbb{C}}$\u7684\u57fa\u5e95\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

1. \u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u5f20\u91cf\u79ef 1.1. \u4e24\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u4e58\u79ef\u7a7a\u95f4 \u5047\u8bbe$V,W$\u662f\u5b9e\u6570… \u7ee7\u7eed\u9605\u8bfb\u5b9e\u5411\u91cf\u7a7a\u95f4\u7684\u590d\u5316<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[218,217,221,220,219],"class_list":["post-1472","post","type-post","status-publish","format-standard","hentry","category-math","tag-fuhua","tag-shixiangliangkongjian","tag-zhangliangji","tag-zhangliangkongjian","tag-fanxingzhi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1472","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1472"}],"version-history":[{"count":4,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1472\/revisions"}],"predecessor-version":[{"id":1476,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1472\/revisions\/1476"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1472"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1472"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1472"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}