Formalization of Two Fixed-Point Algorithms in Hilbert Spaces

Iterative algorithms are fundamental tools for approximating fixed-points of nonexpansive operators in real Hilbert spaces. Among them, Krasnosel’ski\uı–Mann iteration and Halpern iteration are two widely used schemes. In this work, we formalize the convergence of these two fixed-point algorithms in the interactive theorem prover Lean4 based on type dependent theory. To this end, weak convergence and topological properties in the infinite-dimensional real Hilbert space are formalized. Definition and properties of nonexpansive operators are also provided. As a useful tool in convex analysis, we then formalize the Fejér monotone sequence. Building on these foundations, we verify the convergence of both the iteration schemes. Our formalization provides reusable components for machine-checked convergence analysis of fixed-point iterations and theories of convex analysis in real Hilbert spaces. Our code is available at https://github.com/TTony2019/fixed-point-iterations-in-lean.

💡 Research Summary

The paper presents a comprehensive formal verification of two fundamental fixed‑point iteration schemes—Krasnosel’skiĭ–Mann (KM) iteration and Halpern iteration—within the Lean 4 interactive theorem prover. The authors focus on real Hilbert spaces, where the underlying operator (T : H \to H) is assumed to be non‑expansive (i.e., (|Tx - Ty| \le |x-y|) for all (x,y\in H)). Their work proceeds through several tightly connected stages.

First, they develop a formal treatment of the weak topology on a Hilbert space. While mathlib already provides the weak‑star topology (WeakDual) via bilinear pairings, it lacks a direct representation of the weak topology (WeakSpace) and the associated convergence notion. The authors introduce a type synonym WeakSpace R H by applying the flipped canonical pairing between the space and its continuous dual, and define coercions toWeakSpace and ofWeakSpace. Weak convergence of a sequence (x : \mathbb{N}\to H) to a point (p) is expressed as WeakConverge x p, i.e. a Tendsto statement in the filter‑based topology of WeakSpace.

A central result, weakConverge_iff_inner_converge, proves the classical equivalence: a sequence converges weakly to (p) iff the scalar inner products (\langle x_n, y\rangle) converge to (\langle p, y\rangle) for every (y\in H). The proof relies on the Riesz representation theorem, which requires the Hilbert space to be complete (`