Subspace of a Hilbert space with a distinct inner product

Question

I don't really know where to begin with the following question:

Let $ (H_0, \langle \cdot \rangle_0)$ be a closed subspace of $ (H, \langle \cdot \rangle )$ such that norms induced by $ \langle \cdot \rangle, \langle \cdot \rangle_0 $ are equivalent in $ H_0 $. I'm trying to prove that for all $ h \in H $ there exists a unique $ h_0 \in H $ such that $ \forall_{x \in H_0} \langle x,h \rangle = \langle x, h_0 \rangle_0 $

Uniqueness is rather an easy excercise, yet I don't really have any clue what theorem might be useful. The definition of all those notions doesn't seem really helpful to me.

I'd appreciate some help

Answer 1

$x\mapsto\langle x, h\rangle$ is a continous linear functional on $H_0$, therefore, if $H_0, \langle, \rangle_0$ is a Hilbert space, this follows from a well known theorem of Riesz.

(See http://en.wikipedia.org/wiki/Riesz_representation_theorem)