 Solution

The dual space consists of the linear functionals on the original vector space, with addition and scalar multiplication defined pointwise:The dual map acts by function composition with $T$:For $u\in U$, its value is $(T^{\prime }\lambda )(u)=\lambda (Tu)$. The composite is a linear functional because both $T$ and $\lambda$ are linear maps. Moreover $T^{\prime }(a\lambda +b\mu )=a{T}^{\prime }\lambda +b{T}^{\prime }\mu$, so the dual map is itself a linear map. Its direction is reversed because a measurement on the output of $T$ becomes a measurement on its input.