The Kuratowski embedding

Any separable metric space can be embedded isometrically in ℓ^∞(ℕ, ℝ). Any partially defined Lipschitz map into ℓ^∞ can be extended to the whole space.

Any separable metric space can be embedded isometrically in ℓ^∞(ℕ, ℝ)

def KuratowskiEmbedding.embeddingOfSubset {α : Type u} [MetricSpace α] (x : ℕ → α) (a : α) : ↥(lp (fun (i : ℕ) => ℝ) ⊤)

A metric space can be embedded in l^∞(ℝ) via the distances to points in a fixed countable set, if this set is dense. This map is given in kuratowskiEmbedding, without density assumptions.

Equations

• KuratowskiEmbedding.embeddingOfSubset x a = ⟨fun (n : ℕ) => dist a (x n) - dist (x 0) (x n), ⋯⟩

theorem KuratowskiEmbedding.embeddingOfSubset_coe {α : Type u} {n : ℕ} [MetricSpace α] (x : ℕ → α) (a : α) : ↑(KuratowskiEmbedding.embeddingOfSubset x a) n = dist a (x n) - dist (x 0) (x n)

theorem KuratowskiEmbedding.embeddingOfSubset_dist_le {α : Type u} [MetricSpace α] (x : ℕ → α) (a b : α) : dist (KuratowskiEmbedding.embeddingOfSubset x a) (KuratowskiEmbedding.embeddingOfSubset x b) ≤ dist a b

The embedding map is always a semi-contraction.

theorem KuratowskiEmbedding.embeddingOfSubset_isometry {α : Type u} [MetricSpace α] (x : ℕ → α) (H : DenseRange x) : Isometry (KuratowskiEmbedding.embeddingOfSubset x)

When the reference set is dense, the embedding map is an isometry on its image.

theorem KuratowskiEmbedding.exists_isometric_embedding (α : Type u) [MetricSpace α] [TopologicalSpace.SeparableSpace α] : ∃ (f : α → ↥(lp (fun (i : ℕ) => ℝ) ⊤)), Isometry f

Every separable metric space embeds isometrically in ℓ^∞(ℕ).

def kuratowskiEmbedding (α : Type u) [MetricSpace α] [TopologicalSpace.SeparableSpace α] : α → ↥(lp (fun (i : ℕ) => ℝ) ⊤)

The Kuratowski embedding is an isometric embedding of a separable metric space in ℓ^∞(ℕ, ℝ).

Equations

• kuratowskiEmbedding α = Classical.choose ⋯

theorem kuratowskiEmbedding.isometry (α : Type u) [MetricSpace α] [TopologicalSpace.SeparableSpace α] : Isometry (kuratowskiEmbedding α)

The Kuratowski embedding is an isometry. Theorem 2.1 of Assaf Naor, Metric Embeddings and Lipschitz Extensions.

def NonemptyCompacts.kuratowskiEmbedding (α : Type u) [MetricSpace α] [CompactSpace α] [Nonempty α] : TopologicalSpace.NonemptyCompacts ↥(lp (fun (i : ℕ) => ℝ) ⊤)

Version of the Kuratowski embedding for nonempty compacts

Equations

• NonemptyCompacts.kuratowskiEmbedding α = { carrier := Set.range (kuratowskiEmbedding α), isCompact' := ⋯, nonempty' := ⋯ }

theorem LipschitzOnWith.extend_lp_infty {α : Type u} [PseudoMetricSpace α] {s : Set α} {ι : Type u_1} {f : α → ↥(lp (fun (i : ι) => ℝ) ⊤)} {K : NNReal} (hfl : LipschitzOnWith K f s) : ∃ (g : α → ↥(lp (fun (i : ι) => ℝ) ⊤)), LipschitzWith K g ∧ Set.EqOn f g s

A function f : α → ℓ^∞(ι, ℝ) which is K-Lipschitz on a subset s admits a K-Lipschitz extension to the whole space.

Theorem 2.2 of Assaf Naor, Metric Embeddings and Lipschitz Extensions

The same result for the case of a finite type ι is implemented in LipschitzOnWith.extend_pi.