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dae740b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 | import Mathlib
namespace OAI
universe u v
namespace CompactBanach
def DoublingAtMost {B : Type*} [MetricSpace B] (K : Set B) (bound : ℕ) : Prop :=
∀ x : K, ∀ r : ℝ, 0 < r →
∃ centers : Finset K, centers.card ≤ bound ∧
∀ y : K, dist y x < r → ∃ c ∈ centers, dist y c < r / ((2 : ℕ) : ℝ)
def AdmitsBiLipschitzEmbedding {B : Type*} [MetricSpace B] (K : Set B)
(E : Type*) [MetricSpace E] : Prop :=
∃ f : K → E, ∃ a : ℝ, 0 < a ∧ ∃ D : ℝ, 1 ≤ D ∧
∀ x y : K, a * dist x y ≤ dist (f x) (f y) ∧
dist (f x) (f y) ≤ D * a * dist x y
theorem main : ∃ Λ : ℕ, ∀ (B : Type u) [NormedAddCommGroup B] [NormedSpace ℝ B]
[CompleteSpace B], ¬ FiniteDimensional ℝ B →
∃ K : Set B, IsCompact K ∧ DoublingAtMost K Λ ∧
∀ (E : Type v) [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E], ¬ AdmitsBiLipschitzEmbedding K E := by
sorry
end CompactBanach
end OAI
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