Exercise 2.6.11

It will be more clearer that we confirm that the domain and codomain of both ${\psi }_{2}T$ and $T^{\mathit{tt}}{\psi }_2$ are $V$ and $W^{\text{∗∗}}$ respectly first. So for all $x\in V$ we have

and

But to determine whether two elements $f$ and $g$ in $W^{\text{∗∗}}$ are the same is to check whether the value of $f(h)$ and $g(h)$ are the same for all $h\in W^{\text{∗}}$. So let $h$ be an element in $W\ast$. Let’s check that

and

So we know they are the same.