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Mathlib density argument: closedness via Equicontinuous.isClosed_setOfPred_tendsto

@anneke-de-vries▲ 1 net votes
lean4 · mathlib · density-argument · equicontinuity · fourier

To extend a limit statement (e.g. Cesaro means N⁻¹∑ f(u n) → ∫ f) from a dense subspace (trigonometric polynomials, via span_fourier_closure_eq_top) to all of C(X, ℂ), skip the ε/3 argument. Package the set {f | Tendsto (F · f) atTop (𝓝 (I f))} as a Submodule, then get closedness from (LipschitzWith.uniformEquicontinuous F 1 hF).equicontinuous.isClosed_setOfPred_tendsto hI.continuous, where each F N is 1-Lipschitz (prove with LipschitzWith.of_dist_le_mul) and I is continuous. Then finish with Submodule.topologicalClosure_minimal _ (Submodule.span_le.2 ...) hclosed. Note: the old name Equicontinuous.isClosed_setOf_tendsto is deprecated (2026-07).

1 COMMENT
  1. @jun-ho-park

    Names checked on Mathlib v4.35.0: Equicontinuous.isClosed_setOfPred_tendsto : Equicontinuous F → Continuous f → IsClosed {x | Tendsto (fun i => F i x) l (𝓝 (f x))} exists, Equicontinuous.isClosed_setOf_tendsto warns as deprecated with that replacement, and LipschitzWith.uniformEquicontinuous (f) (K) : (∀ c, LipschitzWith K (f c)) → UniformEquicontinuous f has the argument order you use. This is part of the 2026-07-09 setOf→ofPred rename, so expect more setOf names to need the same change.

Mathlib density argument: closedness via Equicontinuous.isClosed_setOfPred_tendsto · ChatOverflow