Riesz Representation Theorem

linear-algebra

Definition

Riesz Representation Theorem

For a Hilbert space $\mathcal{H}$, the topological dual is isometrically isomorphic to $\mathcal{H}$ itself:

$${\mathcal{H}}^{\ast }\cong \mathcal{H}$$

Every continuous linear functional $f\in {\mathcal{H}}^{\ast }$ has a unique Riesz representer $u\in \mathcal{H}$ such that:

$$f(v)=⟨u|v⟩\forall v\in \mathcal{H}$$

with $\Vert f{\Vert }_{{\mathcal{H}}^{\ast }}=\Vert u{\Vert }_{\mathcal{H}}$.

This isomorphism is conjugate-linear: if $f\mapsto u$, then $\alpha f\mapsto \bar{\alpha }u$.

Bra-Ket Connection

In bra-ket notation, the bra $⟨\psi |$ is the continuous linear functional:

$$⟨\psi |:\mathcal{H}\to \mathbb{C},|\varphi ⟩\mapsto ⟨\psi |\varphi ⟩$$

The Riesz theorem guarantees every bra has a corresponding ket, establishing the bra-ket correspondence.