Riesz Representation Theorem

linear-algebra

Definition

Riesz Representation Theorem

For a Hilbert space , the topological dual is isometrically isomorphic to itself:

Every continuous linear functional has a unique Riesz representer such that:

with .

This isomorphism is conjugate-linear: if , then .

Bra-Ket Connection

In bra-ket notation, the bra is the continuous linear functional:

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