The Riesz Representation Theorem for a Real Hilbert Space

Let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$, and let $\ell$ be a bounded linear functional on $H$ with norm $\lVert\ell\rVert$. Then the following hold.

1. (Existence)¶ There is a point $z\in H$ such that $\ell(x)=\langle x,z\rangle$ for every $x\in H$.

2. (Uniqueness)¶ If $z,z'\in H$ satisfy $\langle x,z\rangle=\langle x,z'\rangle$ for every $x\in H$, then $z=z'$.

3. (Norm)¶ If $z\in H$ satisfies $\ell(x)=\langle x,z\rangle$ for every $x\in H$, then $\lVert\ell\rVert=|z|$.