Wikipedia contributors (2026), Wikipedia — https://en.wikipedia.org/wiki/Riesz_representation_theorem
Key takeaways
From the reading, NOTE-861.
- The theorem. For every continuous linear functional φ on a real or complex Hilbert space there is a unique f_φ with φ(x) = ⟨x, f_φ⟩ and ‖f_φ‖ = ‖φ‖. It is the vector in (ker φ)^⊥ with φ(f_φ) = ‖φ‖², and the minimum-norm point of φ^{-1}(‖φ‖²). The map y ↦ ⟨·, y⟩ is an antilinear isometric bijection H → H*; over ℝ it is linear.
- Where completeness enters. Only in H = ker φ ⊕ (ker φ)^⊥. The norm formula ‖⟨·, y⟩‖ = ‖y‖ holds in any inner-product space.
- Constructions. f_φ = conj φ(u) u/‖u‖² for any nonzero u ⊥ ker φ; f_φ = Σ_i conj φ(e_i) e_i over any orthonormal basis, with ‖φ‖² = Σ_i |φ(e_i)|²; on ℂⁿ, conjugate transposition.
- Adjoints. For bounded A : H → Z the adjoint A* is defined by Riesz from the functionals h ↦ ⟨z | Ah⟩, and A* = Φ_H^{-1} ∘ ᵗA ∘ Φ_Z: the transpose, read through the two Riesz maps. Self-adjoint, normal and unitary operators are restated in terms of functionals.
- What it does not cover. Unbounded operators, kernels and RKHS (the word “reproducing” does not occur), and spectral theory.
Standing in the record
Filed on 2026-09-26 at the owner’s request, from a list grouped under the heading Riesz Representation Theorem, and read in full on 2026-10-14 in NOTE-861. It is the record’s reference statement of the theorem; LIT-243 is the gentler proof of the same.
THEORY-009 cites it for “Riesz is what defines the adjoint of a general bounded operator”, which the article supports exactly. The theory’s companion clause, that for integral operators the adjoint is the transpose written down directly, is the theory’s own; the article covers matrices only. The reading files no THEORY: the article is a reference for a standard theorem that THEORY-003 and THEORY-009 already use, and no claim in it is one the record lacks.