The adjoint (transpose) operator

Definition 11

Let \(T : X \mapsto Y\) be a continuous linear map, the transpose / adjoint \(T^* : Y^* \mapsto X^*\) is defined to be the map that sends any functional \(\lambda \in Y^*\) to

\[ T^* \lambda = \lambda T \quad\implies\quad (T^* \lambda)(x) = \lambda(T(x)) \]