The Riesz representation theorem on Hilbert spaces is well known, It asserts we can represent a bounded linear function on a Hilbert space $H$ with an inner product on $H$ and vice-versa.
My question: Given an inner product in $H^*$, say $(a,b)_{H^*}$, can I write it as $$(a,b)_{H^*} = \langle a, f \rangle_{H^*, H}$$ where $f \in H$? This is the RRT applied to the Hilbert space $H^*$ with its dual $H$. I think it works but I never saw it so I should get it clarified.
Best Answer
Well, for an arbitrary inner product on $X:=H^*$ it is not going to work, since then $X^*$ need not be isomorphic to $H$.
On the other hand, the Riesz representation gives a linear isomorphism $H\to H^*$, and if the inner product is defined via this isomorphism, i.e. if $$(\langle x,-\rangle,\langle y,-\rangle)_{H^*}=\langle x,y\rangle_{H} $$ for all $x,y\in H, \ f\in H^*$, then your claim is valid:
Let $a,b\in H^*$ then $a=\langle x,-\rangle$ and $b=\langle y,-\rangle$ for some $y\in H$ by Riesz representation, and $a(y)=\langle x,y\rangle$ so we have $$(a,b)_{H^*} = (a,\langle y,-\rangle)=a(y)=\langle a,y\rangle_{H^*,H}\ .$$