nucleation

LIT-243: Hilbert Spaces and the Riesz Representation Theorem

2 min read

Properties5
StatusActive — read 2026-10-14 (NOTE-854) in full; worth reading as a gentle, complete proof of the Riesz theorem from first principles, done twice (by projection, then by an orthonormal basis), and of the Radon–Nikodym theorem from it. Read it knowing that it is student expository work with typographical slips, that its inner product is conjugate-linear in the first slot, that the representer is Σ conj φ(e_k) e_k, and that Radon–Nikodym is proved for finite measures only.
Version2
Tagsmathematics
Cited as “paper” byNOTE-206 — Adler — Hilbert spaces and the Riesz representation theorem, NOTE-854 — Adler — Hilbert spaces and the Riesz representation theorem
Cited as “source” byTHEORY-629 — Every continuous linear functional on a Hilbert space is the inner product with a unique vector of the same norm, given in any orthonormal basis by the conjugated values of the functional on the basis; completeness is what makes it so, and Riesz's 1907 statement on L² asserted only that a representing function exists

Graph View

Ben Adler (2021), University of Chicago Mathematics REU 2021 (student expository paper) — https://math.uchicago.edu/~may/REU2021/REUPapers/Adler.pdf

Key takeaways

From the reading, NOTE-854.

  • Riesz, by projection (Thm 3.5). With the inner product conjugate-linear in its first slot, every continuous linear φ is v ↦ ⟨u, v⟩ for a unique u, ‖u‖ = ‖φ‖; u = conj φ(w) w for a unit w normal to ker φ. Completeness is used only in the projection theorem.
  • Riesz, by basis (Thm 4.9). u = Σ_k conj φ(e_k) e_k for any orthonormal basis, and ‖φ‖² = Σ_k |φ(e_k)|². The groundwork is an infinite Pythagorean theorem, Bessel’s inequality and the description of a closed span as the square-summable combinations (Thms 4.5–4.8).
  • Radon–Nikodym (Thm 5.1). For finite positive μ, ν with ν ≪ μ, Riesz on L²(ν + μ) gives g with ∫ f(1 − g) dν = ∫ f g dμ, then 0 ≤ g < 1 a.e. and dν/dμ = g/(1 − g), by truncation and monotone convergence. The σ-finite, signed and complex cases are asserted, not proved.
  • No kernels. The paper never mentions kernels or RKHS.

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-854. No THEORY, QUESTION or CLAIM cites it. Readings name it as the record’s gentle proof of Riesz (NOTE-289, NOTE-239) and for the Radon–Nikodym derivative that MINE’s bound is tight at (NOTE-828). It is the companion to the Wikipedia statement, LIT-230. The reading files no THEORY: the results are standard theorems, proved here expositorily. For anything leaned on, cite a textbook (Axler, Rudin, Stein–Shakarchi).

Backlinks