Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Algebra / 2 / a / ii / Solution

The adjoint operator is characterized byFor each fixed $v$, the map $u\mapsto ⟨Tu,v⟩$ is a linear functional. Part (i) gives a representing vector, which we define to be $T^{\ast }v$. Hence such a map exists on every vector of $V$. Part (b) proves that this constructed map is linear.