(1)双対写像が線形写像になることの証明

任意の\(\phi_{1},\phi_{2}\in W^{*},\alpha_{1},\alpha_{2}\in K\)と任意の\(\boldsymbol{v}\in V\)について、
\begin{align*} f^{*}\left(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2}\right)\left(\boldsymbol{v}\right) & =\left(\left(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2}\right)\circ f\right)\left(\boldsymbol{v}\right)\\ & =\left(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2}\right)\left(f\left(\boldsymbol{v}\right)\right)\\ & =\alpha_{1}\phi_{1}\left(f\left(\boldsymbol{v}\right)\right)+\alpha_{2}\phi_{2}\left(f\left(\boldsymbol{v}\right)\right)\\ & =\alpha_{1}\left(\phi_{1}\circ f\right)\left(\boldsymbol{v}\right)+\alpha_{2}\left(\phi_{2}\circ f\right)\left(\boldsymbol{v}\right)\\ & =\left(\alpha_{1}f^{*}\left(\phi_{1}\right)\right)\left(\boldsymbol{v}\right)+\left(\alpha_{2}f^{*}\left(\phi_{2}\right)\right)\left(\boldsymbol{v}\right)\\ & =\left(\alpha_{1}f^{*}\left(\phi_{1}\right)+\alpha_{2}f^{*}\left(\phi_{2}\right)\right)\left(\boldsymbol{v}\right) \end{align*} となり、これは任意の\(\boldsymbol{v}\in V\)について成り立っているので、\(f^{*}\left(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2}\right)=\alpha_{1}f^{*}\left(\phi_{1}\right)+\alpha_{2}f^{*}\left(\phi_{2}\right)\)となり\(f^{*}\)は線形写像となる。