File: Mathlib/Analysis/Normed/Lp/LpEquiv.lean

full_name: Memℓp.all
code: /-- When `α` is `Finite`, every `f : PreLp E p` satisfies `Memℓp f p`. -/
theorem Memℓp.all (f : ∀ i, E i) : Memℓp f p 
goal: Memℓp f p

full_name: Equiv.lpPiLp
code: /-- The canonical `Equiv` between `lp E p ≃ PiLp p E` when `E : α → Type u` with `[Finite α]`. -/
def Equiv.lpPiLp : lp E p ≃ PiLp p E 
goal: ↥(lp E p) ≃ PiLp p E

full_name: coe_equiv_lpPiLp
code: theorem coe_equiv_lpPiLp (f : lp E p) : Equiv.lpPiLp f = ⇑f 
goal: Equiv.lpPiLp f = ↑f

full_name: coe_equiv_lpPiLp_symm
code: theorem coe_equiv_lpPiLp_symm (f : PiLp p E) : (Equiv.lpPiLp.symm f : ∀ i, E i) = f 
goal: ↑(Equiv.lpPiLp.symm f) = f

full_name: AddEquiv.lpPiLp
code: /-- The canonical `AddEquiv` between `lp E p` and `PiLp p E` when `E : α → Type u` with
`[Fintype α]`. -/
def AddEquiv.lpPiLp : lp E p ≃+ PiLp p E 
goal: ↥(lp E p) ≃+ PiLp p E

full_name: coe_addEquiv_lpPiLp
code: theorem coe_addEquiv_lpPiLp  (f : lp E p) : AddEquiv.lpPiLp f = ⇑f 
goal: AddEquiv.lpPiLp f = ↑f

full_name: coe_addEquiv_lpPiLp_symm
code: theorem coe_addEquiv_lpPiLp_symm (f : PiLp p E) :
    (AddEquiv.lpPiLp.symm f : ∀ i, E i) = f 
goal: ↑(AddEquiv.lpPiLp.symm f) = f

full_name: equiv_lpPiLp_norm
code: theorem equiv_lpPiLp_norm [Fintype α] (f : lp E p) : ‖Equiv.lpPiLp f‖ = ‖f‖ 
goal: ‖Equiv.lpPiLp f‖ = ‖f‖

full_name: lpPiLpₗᵢ
code: /-- The canonical `LinearIsometryEquiv` between `lp E p` and `PiLp p E` when `E : α → Type u`
with `[Fintype α]` and `[Fact (1 ≤ p)]`. -/
noncomputable def lpPiLpₗᵢ [Fact (1 ≤ p)] : lp E p ≃ₗᵢ[𝕜] PiLp p E 
goal: ↥(lp E p) ≃ₗᵢ[𝕜] PiLp p E

full_name: coe_lpPiLpₗᵢ
code: theorem coe_lpPiLpₗᵢ [Fact (1 ≤ p)] (f : lp E p) : (lpPiLpₗᵢ E 𝕜 f : ∀ i, E i) = ⇑f 
goal: (lpPiLpₗᵢ E 𝕜) f = ↑f

full_name: coe_lpPiLpₗᵢ_symm
code: theorem coe_lpPiLpₗᵢ_symm [Fact (1 ≤ p)] (f : PiLp p E) :
    ((lpPiLpₗᵢ E 𝕜).symm f : ∀ i, E i) = f 
goal: ↑((lpPiLpₗᵢ E 𝕜).symm f) = f

full_name: AddEquiv.lpBCF
code: /-- The canonical map between `lp (fun _ : α ↦ E) ∞` and `α →ᵇ E` as an `AddEquiv`. -/
noncomputable def AddEquiv.lpBCF : lp (fun _ : α ↦ E) ∞ ≃+ (α →ᵇ E) 
goal: ↥(lp (fun x => E) ⊤) ≃+ BoundedContinuousFunction α E

full_name: coe_addEquiv_lpBCF
code: theorem coe_addEquiv_lpBCF (f : lp (fun _ : α ↦ E) ∞) : (AddEquiv.lpBCF f : α → E) = f 
goal: ⇑(AddEquiv.lpBCF f) = ↑f

full_name: coe_addEquiv_lpBCF_symm
code: theorem coe_addEquiv_lpBCF_symm (f : α →ᵇ E) : (AddEquiv.lpBCF.symm f : α → E) = f 
goal: ↑(AddEquiv.lpBCF.symm f) = ⇑f

full_name: lpBCFₗᵢ
code: /-- The canonical map between `lp (fun _ : α ↦ E) ∞` and `α →ᵇ E` as a `LinearIsometryEquiv`. -/
noncomputable def lpBCFₗᵢ : lp (fun _ : α ↦ E) ∞ ≃ₗᵢ[𝕜] α →ᵇ E 
goal: ↥(lp (fun x => E) ⊤) ≃ₗᵢ[𝕜] BoundedContinuousFunction α E

full_name: coe_lpBCFₗᵢ
code: theorem coe_lpBCFₗᵢ (f : lp (fun _ : α ↦ E) ∞) : (lpBCFₗᵢ E 𝕜 f : α → E) = f 
goal: ⇑((lpBCFₗᵢ E 𝕜) f) = ↑f

full_name: coe_lpBCFₗᵢ_symm
code: theorem coe_lpBCFₗᵢ_symm (f : α →ᵇ E) : ((lpBCFₗᵢ E 𝕜).symm f : α → E) = f 
goal: ↑((lpBCFₗᵢ E 𝕜).symm f) = ⇑f

full_name: RingEquiv.lpBCF
code: /-- The canonical map between `lp (fun _ : α ↦ R) ∞` and `α →ᵇ R` as a `RingEquiv`. -/
noncomputable def RingEquiv.lpBCF : lp (fun _ : α ↦ R) ∞ ≃+* (α →ᵇ R) 
goal: ↥(lp (fun x => R) ⊤) ≃+* BoundedContinuousFunction α R

full_name: coe_ringEquiv_lpBCF
code: theorem coe_ringEquiv_lpBCF (f : lp (fun _ : α ↦ R) ∞) : (RingEquiv.lpBCF R f : α → R) = f 
goal: ⇑((RingEquiv.lpBCF R) f) = ↑f

full_name: coe_ringEquiv_lpBCF_symm
code: theorem coe_ringEquiv_lpBCF_symm (f : α →ᵇ R) : ((RingEquiv.lpBCF R).symm f : α → R) = f 
goal: ↑((RingEquiv.lpBCF R).symm f) = ⇑f

full_name: AlgEquiv.lpBCF
code: /-- The canonical map between `lp (fun _ : α ↦ A) ∞` and `α →ᵇ A` as an `AlgEquiv`. -/
noncomputable def AlgEquiv.lpBCF : lp (fun _ : α ↦ A) ∞ ≃ₐ[𝕜] α →ᵇ A 
goal: ↥(lp (fun x => A) ⊤) ≃ₐ[𝕜] BoundedContinuousFunction α A

full_name: coe_algEquiv_lpBCF
code: theorem coe_algEquiv_lpBCF (f : lp (fun _ : α ↦ A) ∞) : (AlgEquiv.lpBCF α A 𝕜 f : α → A) = f 
goal: ⇑((AlgEquiv.lpBCF α A 𝕜) f) = ↑f

full_name: coe_algEquiv_lpBCF_symm
code: theorem coe_algEquiv_lpBCF_symm (f : α →ᵇ A) : ((AlgEquiv.lpBCF α A 𝕜).symm f : α → A) = f 
goal: ↑((AlgEquiv.lpBCF α A 𝕜).symm f) = ⇑f