I have to show the following:
The composition $T_{H^∗} ◦T_H : H → H^{∗∗}$ is a bijective linear isometry and coincides with the canoncial embedding $J : H → H^{∗∗}$. Here H is an arbitrary $\mathbb K$−Hilbert space and $T_H : H → H^∗$ denotes the map of the theorem of Riesz-Fischer, identifying $H$ with $H^∗$ in a canonical way.
How can I show that the composition coincides with J? That the composition is bijective and linear, is actually clear. But isometry?
We have to show $J h = T_{H^*}(T_H(h))$ for all $h \in H$. This is an equation in $H^{**}$, thus it is sufficient to show $(J h)(h^*) = T_{H^*}(T_H(h))(h^*)$ for all $h \in H$ and $h^* \in H^*$.
By definition of $J$ we have $(J h)(h^*) = h^*(h)$. By definition of $T_{H^*}$, we find $T_{H^*}(T_H(h))(h^*) = (T_H(h), h^*)_{H^*}$. The definition of $T_H$ implies $h^*(h) = (h^*, T_H h)_{H^*}$.
Altogether, this shows $J h = T_{H^*}(T_H(h))$. Since $J$ is a bijective linear isometry, the same holds true for $T_{H^*}(T_H(h))$.