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Mathlib.Topology.MetricSpace.Kuratowski

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 : 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.

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

    The embedding map is always a semi-contraction.

    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 : ) => ) )), 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 : ) => ) )

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

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      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

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        theorem LipschitzOnWith.extend_lp_infty {α : Type u} [PseudoMetricSpace α] {s : Set α} {ι : Type u_1} {f : α(lp (fun (i : ι) => ) )} {K : NNReal} (hfl : LipschitzOnWith K f s) :
        ∃ (g : α(lp (fun (i : ι) => ) )), 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][Naor-2015]

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