Properties of Adjoints
Many of the properties of the adjoint construction follow immediately from the contravariant functoriality of the duality we used in its construction. But they can also be determined from the adjoint relation
For example, if we have transformations 
,w\right\rangle_W&=\left\langle T\left(S(u)\right),w\right\rangle_W\\&=\left\langle S(u),T^*(w)\right\rangle_V\\&=\left\langle u,S^*\left(T^*(w)\right)\right\rangle_U\\&=\left\langle u,\left[S^*T^*\right](w)\right\rangle_W\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Clangle%5Cleft%5BTS%5Cright%5D%28u%29%2Cw%5Cright%5Crangle_W%26%3D%5Cleft%5Clangle+T%5Cleft%28S%28u%29%5Cright%29%2Cw%5Cright%5Crangle_W%5C%5C%26%3D%5Cleft%5Clangle+S%28u%29%2CT%5E%2A%28w%29%5Cright%5Crangle_V%5C%5C%26%3D%5Cleft%5Clangle+u%2CS%5E%2A%5Cleft%28T%5E%2A%28w%29%5Cright%29%5Cright%5Crangle_U%5C%5C%26%3D%5Cleft%5Clangle+u%2C%5Cleft%5BS%5E%2AT%5E%2A%5Cright%5D%28w%29%5Cright%5Crangle_W%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left\langle\left[TS\right](u),w\right\rangle_W&=\left\langle T\left(S(u)\right),w\right\rangle_W\\&=\left\langle S(u),T^*(w)\right\rangle_V\\&=\left\langle u,S^*\left(T^*(w)\right)\right\rangle_U\\&=\left\langle u,\left[S^*T^*\right](w)\right\rangle_W\end{aligned}")
It’s pretty straightforward to see that 
Similarly, it’s easy to show that 
,w\right\rangle_W&=\left\langle T(\lambda v),w\right\rangle_W\\&=\left\langle\lambda v,T^*(w)\right\rangle_V\\&=\left\langle v,\bar{\lambda}T^*(w)\right\rangle_V\\&=\left\langle v,\left[\bar{\lambda}T^*\right](w)\right\rangle_V\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%5Cleft%5Clangle%5Cleft%5B%5Clambda+T%5Cright%5D%28v%29%2Cw%5Cright%5Crangle_W%26%3D%5Cleft%5Clangle+T%28%5Clambda+v%29%2Cw%5Cright%5Crangle_W%5C%5C%26%3D%5Cleft%5Clangle%5Clambda+v%2CT%5E%2A%28w%29%5Cright%5Crangle_V%5C%5C%26%3D%5Cleft%5Clangle+v%2C%5Cbar%7B%5Clambda%7DT%5E%2A%28w%29%5Cright%5Crangle_V%5C%5C%26%3D%5Cleft%5Clangle+v%2C%5Cleft%5B%5Cbar%7B%5Clambda%7DT%5E%2A%5Cright%5D%28w%29%5Cright%5Crangle_V%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\displaystyle\begin{aligned}\left\langle\left[\lambda T\right](v),w\right\rangle_W&=\left\langle T(\lambda v),w\right\rangle_W\\&=\left\langle\lambda v,T^*(w)\right\rangle_V\\&=\left\langle v,\bar{\lambda}T^*(w)\right\rangle_V\\&=\left\langle v,\left[\bar{\lambda}T^*\right](w)\right\rangle_V\end{aligned}")
Now if we restrict our focus to endomorphisms of a single vector space 
In a way, then, the adjoint behaves sort of like the complex conjugate itself does for the algebra of complex numbers (over the complex numbers we don’t notice the order of multiplication, but work with me here, people). This analogy goes pretty far, as we’ll see.
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