Suppose we have a Hilbert Space $X$, with a Hilbert subspace $H \subset X$.
For each $f \in X^*$, we can restrict it on $H$, as a consequence we have $X^* \subset H^*$; and the two dual space here are not equivalent in general.
But as a corollary of Rietz representation theorem, there is a isomorphism between $X$ and $X^*$, $H$ and $H^*$. In other words, we can treat the element $h \in H$ as an element in $H^*$, by using $h(u) = (h|u)$. We can prove that this is the form of any element in $H^*$;
In all, we have the seemingly controversial result as $$ H \subset X = X^* \subset H^* = H$$
What's wrong with this? Have I misunderstood something?
The morphism $X^*\rightarrow H^*$ is not always injective. Take $x$ orthogonal to $H$, and $f(x)=\langle x,.\rangle$. The restriction of $f_x$ to $H$ is zero. So $X^*$ is not imbedded in $H^*$.