# 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} [] (x : α) (a : α) :
(lp (fun (i : ) => ) )

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
Instances For
theorem KuratowskiEmbedding.embeddingOfSubset_coe {α : Type u} {n : } [] (x : α) (a : α) :
= dist a (x n) - dist (x 0) (x n)
theorem KuratowskiEmbedding.embeddingOfSubset_dist_le {α : Type u} [] (x : α) (a : α) (b : α) :

The embedding map is always a semi-contraction.

theorem KuratowskiEmbedding.embeddingOfSubset_isometry {α : Type u} [] (x : α) (H : ) :

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

theorem KuratowskiEmbedding.exists_isometric_embedding (α : Type u) [] :
∃ (f : α(lp (fun (i : ) => ) )),

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

def kuratowskiEmbedding (α : Type u) [] :
α(lp (fun (i : ) => ) )

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

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Instances For
theorem kuratowskiEmbedding.isometry (α : Type u) [] :

The Kuratowski embedding is an isometry. Theorem 2.1 of [Assaf Naor, Metric Embeddings and Lipschitz Extensions][Naor-2015].

Version of the Kuratowski embedding for nonempty compacts

Equations
• = { carrier := , isCompact' := , nonempty' := }
Instances For
theorem LipschitzOnWith.extend_lp_infty {α : Type u} {s : Set α} {ι : Type u_1} {f : α(lp (fun (i : ι) => ) )} {K : NNReal} (hfl : ) :
∃ (g : α(lp (fun (i : ι) => ) )), 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][Naor-2015]

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