Reisz Representation Theorem

Specific Case:

Think of vector space \(\mathbb{R}^n\). Every linear function in that vector space can be expressed as a dot product with some vector \(x\) in that space. i.e.

for any linear \(\phi: V \rightarrow \mathbb{R}\) there is a \(x\) such that \(\phi(y) = x \cdot y\).

Reisz Representation Theorem is a generalization of this.

Definition:

If we take a bounded linear functional \(\phi : H \rightarrow \mathbb{R}\) in Hilbert Space \(H\) then, there exits some vector \(x \in H\) such that function values of the functional are given by scalar product with that vector \(x\):

\[ \phi(y) = \langle x, y \rangle \]

The space of all linear functional is called the Dual (conjugate) vector space denoted by \(H^*\).

In other words,

For \(\phi \in H^*\) there exits \(f_{\phi} \in H\) called Reisz Representation of \(\phi\) such that

\[ \phi(y) = \langle f_{\phi}, y \rangle \]

Properties:

• The operator norm of the functional \(\phi\) is equal to Hilbert space norm of the vector \(x\): \(||\phi|| = ||x||\)
• \(f_\phi\) is a unique vector in \((\ker \phi)^\bot\) satisfying \(\phi(f_\phi) = ||\phi||^2\)