is it riesz representation theorem

Question

If we have the mapping: x $\to$ f(x) from H into $\mathbb{R}$ is an isomorphism ,where H is an hilbert space .\ my question is :According to what we have the right to write that there exists a unique y $\in$ H such that $f(x)=<x,y>$.Is it by riesz theorem and how?

Answer 1

Yes this is Riesz representation theorem

Let $H^\star$ be the space of linear functionals (in particular $f\in H^\star$)

For $y\in H$ we can define a map $\varphi_y\in H^\star$ by $\varphi_y(x)=\left<x,y\right>$.

By Riesz representation theorem there exist an isometric isomorphism $H\rightarrow H^\star$ taking $y\in H$ to the functional $\varphi_y$.

In particular the isomorphism is a bijection and so for $f\in H^\star$ there exists $y$ such that $f=\varphi_y$.