File size: 2,450 Bytes
9655094 | 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 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Analysis.C0Absorption.Operators
namespace OAI
open Set Filter Topology
open scoped NNReal BigOperators ZeroAtInfty
open NormedSpace
namespace C0Absorption
noncomputable section
open Set Filter Topology
open scoped NNReal BigOperators
section LocalOnto
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
theorem range_mem_nhds_of_approx {f : E → F} {a : E} (S : E ≃L[ℝ] F)
(s : Set E) (hs : s∈nhds a) (c : ℝ≥0)
(hc : c<‖(S.symm : F →L[ℝ] E)‖₊⁻¹) (ha : ApproximatesLinearOn f (S : E →L[ℝ] F) s c) :
Set.range f∈nhds (f a) := by
have hh := ha.image_mem_nhds S.toNonlinearRightInverse hs (Or.inr hc)
exact mem_of_superset hh (Set.image_subset_range f s)
theorem surjective_of_punctured_local_onto {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
[Nontrivial E]
(f : E → F) {c C : ℝ≥0} (hf : AntilipschitzWith c f) (hLip : LipschitzWith C f)
(hzero : f 0=0) (hconn : IsPreconnected ({0}ᶜ : Set F))
(hlocal : ∀ x≠0, Set.range f∈nhds (f x)) : Function.Surjective f := by
let K : Set F := {0}ᶜ
let A : Set K := Subtype.val ⁻¹' Set.range f
let : PreconnectedSpace K := Subtype.preconnectedSpace hconn
have hclosed : IsClosed A := (hf.isClosed_range hLip.uniformContinuous).preimage continuous_subtype_val
have hopen : IsOpen A := by
rw [isOpen_iff_mem_nhds]
intro y hy
obtain ⟨x,hx⟩ := hy
have hx0 : x≠0 := by
intro hx0
have he : y.val=0 := by rw [← hx,hx0,hzero]
exact y.property (by simpa only [Set.mem_singleton_iff] using he)
have hn : Set.range f∈nhds y.val := by rw [← hx]; exact hlocal x hx0
exact continuous_subtype_val.continuousAt hn
have hne : A.Nonempty := by
obtain ⟨x,hx⟩ := exists_ne (0 : E)
have hx' : f x≠0 := by intro he; exact hx (hf.injective (he.trans hzero.symm))
refine ⟨⟨f x,?_⟩,x,rfl⟩
simpa only [K,Set.mem_compl_iff,Set.mem_singleton_iff] using hx'
have heq : A=Set.univ := IsClopen.eq_univ ⟨hclosed,hopen⟩ hne
intro y
by_cases hy : y=0
· exact ⟨0,hzero.trans hy.symm⟩
· let yy : K := ⟨y,by simpa only [K,Set.mem_compl_iff,Set.mem_singleton_iff] using hy⟩
have hh : yy∈A := by rw [heq]; trivial
exact hh
end LocalOnto
end
end C0Absorption
end OAI
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