📂Hilbert Space

Proof of Liouville's Theorem

Theorem[^1]

Let $\left( H, \left\langle \cdot,\cdot \right\rangle \right)$ be a Hilbert space. For linear functionals $f \in H^{ \ast }$ and $\mathbf{x} \in H$ satisfying $f ( \mathbf{x} ) = \left\langle \mathbf{x} , \mathbf{w} \right\rangle$ and $\| f \|_{H^{\ast}} = \| \mathbf{w} \|_{H}$, there exists a unique $\mathbf{w} \in H$.