Riesz Representation Theorem (Hilbert Spaces)

Theorem
Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$.

Then there is a unique $h_0 \in H$ such that


 * $\forall h \in H: Lh = \left\langle{h, h_0}\right\rangle$

Corollary
In the notation above, the norm of $L$ satisfies


 * $\left\|{L}\right\| = \left\|{h_0}\right\|$