File: Mathlib/Analysis/InnerProductSpace/Dual.lean

full_name: InnerProductSpace.«_aux_Mathlib_Analysis_InnerProductSpace_Dual___macroRules_InnerProductSpace_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
goal: Lean.Macro

full_name: InnerProductSpace.«_aux_Mathlib_Analysis_InnerProductSpace_Dual___macroRules_InnerProductSpace_term_†_1»
code: local postfix:90 "†" => starRingEnd _
goal: Lean.Macro

full_name: InnerProductSpace.toDualMap
code: /-- An element `x` of an inner product space `E` induces an element of the dual space `Dual 𝕜 E`,
the map `fun y => ⟪x, y⟫`; moreover this operation is a conjugate-linear isometric embedding of `E`
into `Dual 𝕜 E`.
If `E` is complete, this operation is surjective, hence a conjugate-linear isometric equivalence;
see `toDual`.
-/
def toDualMap : E →ₗᵢ⋆[𝕜] NormedSpace.Dual 𝕜 E 
goal: E →ₗᵢ⋆[𝕜] NormedSpace.Dual 𝕜 E

full_name: InnerProductSpace.toDualMap_apply
code: @[simp]
theorem toDualMap_apply {x y : E} : toDualMap 𝕜 E x y = ⟪x, y⟫ 
goal: ((InnerProductSpace.toDualMap 𝕜 E) x) y = ⟪x, y⟫_𝕜

full_name: InnerProductSpace.innerSL_norm
code: theorem innerSL_norm [Nontrivial E] : ‖(innerSL 𝕜 : E →L⋆[𝕜] E →L[𝕜] 𝕜)‖ = 1 
goal: ‖innerSL 𝕜‖ = 1

full_name: InnerProductSpace.ext_inner_left_basis
code: theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
    (h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y 
goal: x = y

full_name: InnerProductSpace.ext_inner_right_basis
code: theorem ext_inner_right_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
    (h : ∀ i : ι, ⟪x, b i⟫ = ⟪y, b i⟫) : x = y 
goal: x = y

full_name: InnerProductSpace.toDual
code: /-- Fréchet-Riesz representation: any `ℓ` in the dual of a Hilbert space `E` is of the form
`fun u => ⟪y, u⟫` for some `y : E`, i.e. `toDualMap` is surjective.
-/
def toDual : E ≃ₗᵢ⋆[𝕜] NormedSpace.Dual 𝕜 E 
goal: E ≃ₗᵢ⋆[𝕜] NormedSpace.Dual 𝕜 E

full_name: InnerProductSpace.toDual_apply
code: @[simp]
theorem toDual_apply {x y : E} : toDual 𝕜 E x y = ⟪x, y⟫ 
goal: ((InnerProductSpace.toDual 𝕜 E) x) y = ⟪x, y⟫_𝕜

full_name: InnerProductSpace.toDual_symm_apply
code: @[simp]
theorem toDual_symm_apply {x : E} {y : NormedSpace.Dual 𝕜 E} : ⟪(toDual 𝕜 E).symm y, x⟫ = y x 
goal: ⟪(InnerProductSpace.toDual 𝕜 E).symm y, x⟫_𝕜 = y x

full_name: InnerProductSpace.continuousLinearMapOfBilin
code: /-- Maps a bounded sesquilinear form to its continuous linear map,
given by interpreting the form as a map `B : E →L⋆[𝕜] NormedSpace.Dual 𝕜 E`
and dualizing the result using `toDual`.
-/
def continuousLinearMapOfBilin (B : E →L⋆[𝕜] E →L[𝕜] 𝕜) : E →L[𝕜] E 
goal: E →L[𝕜] E

full_name: InnerProductSpace.«_aux_Mathlib_Analysis_InnerProductSpace_Dual___macroRules_InnerProductSpace_term_♯_1»
code: local postfix:1024 "♯" => continuousLinearMapOfBilin
goal: Lean.Macro

full_name: InnerProductSpace._aux_Mathlib_Analysis_InnerProductSpace_Dual___unexpand_InnerProductSpace_continuousLinearMapOfBilin_1
code: local postfix:1024 "♯" => continuousLinearMapOfBilin
goal: Lean.PrettyPrinter.Unexpander

full_name: InnerProductSpace.continuousLinearMapOfBilin_apply
code: @[simp]
theorem continuousLinearMapOfBilin_apply (v w : E) : ⟪B♯ v, w⟫ = B v w 
goal: ⟪(InnerProductSpace.continuousLinearMapOfBilin B) v, w⟫_𝕜 = (B v) w

full_name: InnerProductSpace.unique_continuousLinearMapOfBilin
code: theorem unique_continuousLinearMapOfBilin {v f : E} (is_lax_milgram : ∀ w, ⟪f, w⟫ = B v w) :
    f = B♯ v 
goal: f = (InnerProductSpace.continuousLinearMapOfBilin B) v