Source
Adler (2021), “Hilbert Spaces and the Riesz Representation Theorem”, University of Chicago REU paper, LIT-243, read in full in NOTE-854. Riesz (1907), “Sur une espèce de géométrie analytique des systèmes de fonctions sommables”, Comptes rendus 144:1409–1411, LIT-326, read in full in NOTE-289.
What was actually shown
The theorem (LIT-243, Thm 3.5). With the inner product conjugate-linear in its first slot, for every continuous linear φ on a Hilbert space V there is a unique u with φ(v) = ⟨u, v⟩ for all v, and ‖u‖ = ‖φ‖. The proof takes a unit w normal to ker φ and sets u = conj φ(w) w. Completeness enters once, in the Hilbert projection theorem (Thm 2.2), where a minimising sequence must converge.
The basis formula (Thm 4.9). For any orthonormal basis {e_k}, u = Σ_k conj φ(e_k) e_k and ‖φ‖² = Σ_k |φ(e_k)|². The groundwork is Bessel’s inequality and the closed span as the square-summable combinations (Thms 4.5–4.8). This is the finite-dimensional dual-basis computation of §3, carried to infinite dimensions.
A consequence (Thm 5.1). Applied to f ↦ ∫ f dν on L²(ν + μ), the theorem gives the Radon–Nikodym derivative dν/dμ = g/(1 − g) for finite positive measures with ν ≪ μ.
The first statement (LIT-326, p. 1410). Riesz states that for every operation U on square-summable functions that is linear and continuous for convergence in mean, there is a function k with U(f) = ∫ f k. He calls it an immediate consequence of his existence theorem, that every square-summable coordinate sequence is the image of a square-integrable function (the Riesz–Fischer theorem), and gives no argument.
What could have come out otherwise. On an incomplete inner-product space the projection step fails, and a continuous functional need not be represented by any vector of the space.
What this does not say
- Anything about kernels, RKHS or factorisation. Neither paper mentions them (NOTE-854). A “unique factorization” argument must bring its own uniqueness: of the representer here, or of a representation up to an orthogonal transformation (THEORY-004). Riesz 1907 states no uniqueness at all (NOTE-289).
- That the 1907 note proves the theorem, or proves it for Hilbert space. It asserts existence of k on L² only, with no uniqueness, no norm identity and an unspecified domain. Fréchet’s note in the same issue states it simultaneously.
- That the representer is Σ φ(e_k) e_k. The conjugate matters on complex spaces; LIT-243’s seeded summary dropped it (NOTE-854).
- That the bracket of Dirac notation is an inner product everywhere. It is only where the theorem holds. On a Banach space bras and kets can be written, but the bracket is only a functional applied to a vector (LIT-237, NOTE-855).
- Radon–Nikodym in general. LIT-243 proves it for finite measures; the σ-finite, signed and complex cases are asserted.
Connections
- THEORY-009 holds that spectral self-supervised learning gets self-adjointness from symmetry of the pair distribution, not from this theorem. THEORY-003 is the one place in that family where it is invoked: kernel mean embeddings exist by it. THEORY-314’s representer theorem uses the reproducing property, not this theorem.
- LIT-230, the Wikipedia statement, gives the same theorem in the opposite convention.