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a89d282 | 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 36 37 38 | import OAI.Probability.InvariantIsing.Haar.HaarFrobeniusGeometry
namespace OAI
/-! Frobenius-Lipschitz observables are continuous in the matrix-group topology. -/
noncomputable section
open Matrix Filter
open scoped Topology
namespace InvariantIsing
lemma continuous_frobeniusDistance_right {N : ℕ} (U : SpecialOrthogonal N) :
Continuous (fun V => frobeniusDistance V U) := by
have he (i j : Fin N) : Continuous (fun V : SpecialOrthogonal N =>
(V : Matrix (Fin N) (Fin N) ℝ) i j) :=
(continuous_apply j).comp ((continuous_apply i).comp continuous_subtype_val)
unfold frobeniusDistance
apply Real.continuous_sqrt.comp
exact continuous_finsetSum _ fun i _ => continuous_finsetSum _ fun j _ =>
((he i j).sub continuous_const).pow 2
lemma frobeniusDistance_self {N : ℕ} (U : SpecialOrthogonal N) : frobeniusDistance U U = 0 := by
simp [frobeniusDistance]
theorem continuous_of_frobenius_lipschitz {N : ℕ} (f : SpecialOrthogonal N → ℝ) (L : ℝ)
(hf : ∀ U V, |f U-f V| ≤ L*frobeniusDistance U V) : Continuous f := by
apply continuous_iff_continuousAt.mpr
intro U
apply tendsto_iff_norm_sub_tendsto_zero.mpr
apply squeeze_zero (fun V => norm_nonneg _) (fun V => by simpa only [Real.norm_eq_abs] using hf V U)
have hh := ((continuous_frobeniusDistance_right U).tendsto U).const_mul L
simpa only [frobeniusDistance_self,mul_zero] using hh
end InvariantIsing
end
end OAI
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