双線型写像
V × V ⟶ M R ( v , w ) ↦ v ′ M w M i j = M ( b i , b j ) M ∈ V ∗ ⊗ V ∗ M = M s t β s ⊗ β t V × V ∗ ⟶ M R ( v , f ) ↦ v ′ M f ′ M i j = M ( b i , β j ) M ∈ V ∗ ⊗ V M = M s t β s ⊗ b t M s t = M s u g u t M G − 1 V ∗ × V ⟶ M R ( f , w ) ↦ f M w M j i = M ( β i , b j ) M ∈ V ⊗ V ∗ M = M t s b s ⊗ β t M t s = g s u M u t G − 1 M V ∗ × V ∗ ⟶ M R ( f , g ) ↦ f M g ′ M i j = M ( β i , β j ) M ∈ V ⊗ V M = M s t b s ⊗ b t M s t = g s u M u v g v t G − 1 M G − 1 {\displaystyle {\begin{matrix}{\begin{aligned}&V\times V{\overset {M}{\longrightarrow }}\mathbb {R} \\&(v,w)\mapsto v'Mw\\&M_{ij}=M(b_{i},b_{j})\\&M\in V^{*}\otimes V^{*}\\&M=M_{st}\beta ^{s}\otimes \beta ^{t}\\~\\~\end{aligned}}&{\begin{aligned}&V\times V^{*}{\overset {M}{\longrightarrow }}\mathbb {R} \\&(v,f)\mapsto v'Mf'\\&M_{i}^{j}=M(b_{i},\beta ^{j})\\&M\in V^{*}\otimes V\\&M=M_{s}^{t}\beta ^{s}\otimes b_{t}\\&M_{s}^{t}=M_{su}g^{ut}\\&MG^{-1}\end{aligned}}\\{\begin{aligned}&V^{*}\times V{\overset {M}{\longrightarrow }}\mathbb {R} \\&(f,w)\mapsto fMw\\&M_{j}^{i}=M(\beta ^{i},b_{j})\\&M\in V\otimes V^{*}\\&M=M_{t}^{s}b_{s}\otimes \beta ^{t}\\&M_{t}^{s}=g^{su}M_{ut}\\&G^{-1}M\end{aligned}}&{\begin{aligned}&V^{*}\times V^{*}{\overset {M}{\longrightarrow }}\mathbb {R} \\&(f,g)\mapsto fMg'\\&M^{ij}=M(\beta ^{i},\beta ^{j})\\&M\in V\otimes V\\&M=M^{st}b_{s}\otimes b_{t}\\&M^{st}=g^{su}M_{uv}g^{vt}\\&G^{-1}MG^{-1}\end{aligned}}\end{matrix}}}