Riesz Representation Theorem

Not to be confused with the other (TODO link) theorem with this name. Another name for the present one is Fréchet–Riesz Representation Theorem.

For any inner product space $X$, taking the inner product with a fixed vector $y$, that is, the mapping defined by $x\mapsto ⟨y,x⟩$, is a bounded linear functional on $X$. The Riesz Representation Theorem states that if $X$ is a Hilbert space, then every bounded linear functional is of this form.

Riesz Representation Theorem

For every bounded linear functional $f$ on a Hibert space $\mathcal{H}$ there is a unique vector $y\in \mathcal{H}$ such that

$$f(x)=⟨y,x⟩\forall x\in \mathcal{H}.$$

Moreover, $\Vert f\Vert =\Vert y\Vert$.

Under these circumstances, the functional $f$ is said to be represented by the vector $y$.

The mapping ${\mathcal{H}}^{\prime }\to \mathcal{H}$, which maps a bounded linear functional on $\mathcal{H}$ to its unique representing vector in $\mathcal{H}$, is an isometric anti-isomorphism.

Here, anti-isomorphism means anti-linear bijection. In the real case, the mapping is an isometric isomorphism.

The dual space of a Hilbert space is a Hilbert space.

Every Hilbert space is reflexive.

Extension of bounded linear functionals defined on a subspace