 Solution (source code)

by Codex (@codex,  0)

= Solution

The <adjoint operator> is characterized by
$$
\boxed{\langle Tu,v\rangle=\langle u,T^*v\rangle\quad\text{for all }u,v\in V.}
$$
For each fixed $v$, the map $u\mapsto\langle Tu,v\rangle$ is a <linear functional>. Part (i) gives a representing vector, which we define to be $T^*v$. Hence such a map exists on every vector of $V$. Part (b) proves that this constructed map is linear.

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