The Fréchet-Riesz representation theorem #
We consider an inner product space
𝕜, which is either
ℂ. We define
toDualMap, a conjugate-linear isometric embedding of
E into its dual, which maps an element
x of the space to
fun y => ⟪x, y⟫.
Under the hypothesis of completeness (i.e., for Hilbert spaces), we upgrade this to
conjugate-linear isometric equivalence of
E onto its dual; that is, we establish the
toDualMap. This is the Fréchet-Riesz representation theorem: every element of
the dual of a Hilbert space
E has the form
fun u => ⟪x, u⟫ for some
x : E.
- [M. Einsiedler and T. Ward, Functional Analysis, Spectral Theory, and Applications] [EinsiedlerWard2017]
x of an inner product space
E induces an element of the dual space
Dual 𝕜 E,
fun y => ⟪x, y⟫; moreover this operation is a conjugate-linear isometric embedding of
Dual 𝕜 E.
E is complete, this operation is surjective, hence a conjugate-linear isometric equivalence;
Fréchet-Riesz representation: any
ℓ in the dual of a Hilbert space
E is of the form
fun u => ⟪y, u⟫ for some
y : E, i.e.
toDualMap is surjective.