File: Mathlib/MeasureTheory/Function/L2Space.lean

full_name: MeasureTheory.Memℒp.integrable_sq
code: theorem Memℒp.integrable_sq {f : α → ℝ} (h : Memℒp f 2 μ) : Integrable (fun x => f x ^ 2) μ 
goal: MeasureTheory.Integrable (fun x => f x ^ 2) μ

full_name: MeasureTheory.memℒp_two_iff_integrable_sq_norm
code: theorem memℒp_two_iff_integrable_sq_norm {f : α → F} (hf : AEStronglyMeasurable f μ) :
    Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ 
goal: MeasureTheory.Memℒp f 2 μ ↔ MeasureTheory.Integrable (fun x => ‖f x‖ ^ 2) μ

full_name: MeasureTheory.memℒp_two_iff_integrable_sq
code: theorem memℒp_two_iff_integrable_sq {f : α → ℝ} (hf : AEStronglyMeasurable f μ) :
    Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ 
goal: MeasureTheory.Memℒp f 2 μ ↔ MeasureTheory.Integrable (fun x => f x ^ 2) μ

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_L2Space___macroRules_MeasureTheory_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
goal: Lean.Macro

full_name: MeasureTheory.Memℒp.const_inner
code: theorem Memℒp.const_inner (c : E) {f : α → E} (hf : Memℒp f p μ) : Memℒp (fun a => ⟪c, f a⟫) p μ 
goal: MeasureTheory.Memℒp (fun a => ⟪c, f a⟫_𝕜) p μ

full_name: MeasureTheory.Memℒp.inner_const
code: theorem Memℒp.inner_const {f : α → E} (hf : Memℒp f p μ) (c : E) : Memℒp (fun a => ⟪f a, c⟫) p μ 
goal: MeasureTheory.Memℒp (fun a => ⟪f a, c⟫_𝕜) p μ

full_name: MeasureTheory.Integrable.const_inner
code: theorem Integrable.const_inner (c : E) (hf : Integrable f μ) :
    Integrable (fun x => ⟪c, f x⟫) μ 
goal: MeasureTheory.Integrable (fun x => ⟪c, f x⟫_𝕜) μ

full_name: MeasureTheory.Integrable.inner_const
code: theorem Integrable.inner_const (hf : Integrable f μ) (c : E) :
    Integrable (fun x => ⟪f x, c⟫) μ 
goal: MeasureTheory.Integrable (fun x => ⟪f x, c⟫_𝕜) μ

full_name: integral_inner
code: theorem _root_.integral_inner {f : α → E} (hf : Integrable f μ) (c : E) :
    ∫ x, ⟪c, f x⟫ ∂μ = ⟪c, ∫ x, f x ∂μ⟫ 
goal: ∫ (x : α), ⟪c, f x⟫_𝕜 ∂μ = ⟪c, ∫ (x : α), f x ∂μ⟫_𝕜

full_name: integral_eq_zero_of_forall_integral_inner_eq_zero
code: theorem _root_.integral_eq_zero_of_forall_integral_inner_eq_zero (f : α → E) (hf : Integrable f μ)
    (hf_int : ∀ c : E, ∫ x, ⟪c, f x⟫ ∂μ = 0) : ∫ x, f x ∂μ = 0 
goal: ∫ (x : α), f x ∂μ = 0

full_name: MeasureTheory.L2.«_aux_Mathlib_MeasureTheory_Function_L2Space___macroRules_MeasureTheory_L2_term⟪_,_⟫_1»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
goal: Lean.Macro

full_name: MeasureTheory.L2.eLpNorm_rpow_two_norm_lt_top
code: theorem eLpNorm_rpow_two_norm_lt_top (f : Lp F 2 μ) :
    eLpNorm (fun x => ‖f x‖ ^ (2 : ℝ)) 1 μ < ∞ 
goal: MeasureTheory.eLpNorm (fun x => ‖↑↑f x‖ ^ 2) 1 μ < ⊤

full_name: MeasureTheory.L2.eLpNorm_inner_lt_top
code: theorem eLpNorm_inner_lt_top (f g : α →₂[μ] E) : eLpNorm (fun x : α => ⟪f x, g x⟫) 1 μ < ∞ 
goal: MeasureTheory.eLpNorm (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) 1 μ < ⊤

full_name: MeasureTheory.L2.instInnerSubtypeAEEqFunMemAddSubgroupLpOfNatENNReal
code: instance : Inner 𝕜 (α →₂[μ] E) 
goal: Inner 𝕜 ↥(MeasureTheory.Lp E 2 μ)

full_name: MeasureTheory.L2.inner_def
code: theorem inner_def (f g : α →₂[μ] E) : ⟪f, g⟫ = ∫ a : α, ⟪f a, g a⟫ ∂μ 
goal: ⟪f, g⟫_𝕜 = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫_𝕜 ∂μ

full_name: MeasureTheory.L2.integral_inner_eq_sq_eLpNorm
code: theorem integral_inner_eq_sq_eLpNorm (f : α →₂[μ] E) :
    ∫ a, ⟪f a, f a⟫ ∂μ = ENNReal.toReal (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ (2 : ℝ) ∂μ) 
goal: ∫ (a : α), ⟪↑↑f a, ↑↑f a⟫_𝕜 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal

full_name: _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.norm_sq_eq_inner'
code: private theorem norm_sq_eq_inner' (f : α →₂[μ] E) : ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫ 
goal: ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫_𝕜

full_name: MeasureTheory.L2.mem_L1_inner
code: theorem mem_L1_inner (f g : α →₂[μ] E) :
    AEEqFun.mk (fun x => ⟪f x, g x⟫)
        ((Lp.aestronglyMeasurable f).inner (Lp.aestronglyMeasurable g)) ∈
      Lp 𝕜 1 μ 
goal: MeasureTheory.AEEqFun.mk (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) ⋯ ∈ MeasureTheory.Lp 𝕜 1 μ

full_name: MeasureTheory.L2.integrable_inner
code: theorem integrable_inner (f g : α →₂[μ] E) : Integrable (fun x : α => ⟪f x, g x⟫) μ 
goal: MeasureTheory.Integrable (fun x => ⟪↑↑f x, ↑↑g x⟫_𝕜) μ

full_name: _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.add_left'
code: private theorem add_left' (f f' g : α →₂[μ] E) : ⟪f + f', g⟫ = inner f g + inner f' g 
goal: ⟪f + f', g⟫_𝕜 = ⟪f, g⟫_𝕜 + ⟪f', g⟫_𝕜

full_name: _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.smul_left'
code: private theorem smul_left' (f g : α →₂[μ] E) (r : 𝕜) : ⟪r • f, g⟫ = conj r * inner f g 
goal: ⟪r • f, g⟫_𝕜 = (starRingEnd 𝕜) r * ⟪f, g⟫_𝕜

full_name: MeasureTheory.L2.innerProductSpace
code: instance innerProductSpace : InnerProductSpace 𝕜 (α →₂[μ] E) 
goal: InnerProductSpace 𝕜 ↥(MeasureTheory.Lp E 2 μ)

full_name: MeasureTheory.L2.inner_indicatorConstLp_eq_setIntegral_inner
code: /-- The inner product in `L2` of the indicator of a set `indicatorConstLp 2 hs hμs c` and `f` is
equal to the integral of the inner product over `s`: `∫ x in s, ⟪c, f x⟫ ∂μ`. -/
theorem inner_indicatorConstLp_eq_setIntegral_inner (f : Lp E 2 μ) (hs : MeasurableSet s) (c : E)
    (hμs : μ s ≠ ∞) : (⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ∫ x in s, ⟪c, f x⟫ ∂μ 
goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs c, f⟫_𝕜 = ∫ (x : α) in s, ⟪c, ↑↑f x⟫_𝕜 ∂μ

full_name: MeasureTheory.L2.inner_indicatorConstLp_eq_inner_setIntegral
code: /-- The inner product in `L2` of the indicator of a set `indicatorConstLp 2 hs hμs c` and `f` is
equal to the inner product of the constant `c` and the integral of `f` over `s`. -/
theorem inner_indicatorConstLp_eq_inner_setIntegral [CompleteSpace E] [NormedSpace ℝ E]
    (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) (f : Lp E 2 μ) :
    (⟪indicatorConstLp 2 hs hμs c, f⟫ : 𝕜) = ⟪c, ∫ x in s, f x ∂μ⟫ 
goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs c, f⟫_𝕜 = ⟪c, ∫ (x : α) in s, ↑↑f x ∂μ⟫_𝕜

full_name: MeasureTheory.L2.inner_indicatorConstLp_one
code: /-- The inner product in `L2` of the indicator of a set `indicatorConstLp 2 hs hμs (1 : 𝕜)` and
a real or complex function `f` is equal to the integral of `f` over `s`. -/
theorem inner_indicatorConstLp_one (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (f : Lp 𝕜 2 μ) :
    ⟪indicatorConstLp 2 hs hμs (1 : 𝕜), f⟫ = ∫ x in s, f x ∂μ 
goal: ⟪MeasureTheory.indicatorConstLp 2 hs hμs 1, f⟫_𝕜 = ∫ (x : α) in s, ↑↑f x ∂μ

full_name: MeasureTheory.«_aux_Mathlib_MeasureTheory_Function_L2Space___macroRules_MeasureTheory_term⟪_,_⟫_2»
code: local notation "⟪" x ", " y "⟫" => @inner 𝕜 (α →₂[μ] 𝕜) _ x y
goal: Lean.Macro

full_name: MeasureTheory.BoundedContinuousFunction.inner_toLp
code: /-- For bounded continuous functions `f`, `g` on a finite-measure topological space `α`, the L^2
inner product is the integral of their pointwise inner product. -/
theorem BoundedContinuousFunction.inner_toLp (f g : α →ᵇ 𝕜) :
    ⟪BoundedContinuousFunction.toLp (E := 𝕜) 2 μ 𝕜 f,
        BoundedContinuousFunction.toLp (E := 𝕜) 2 μ 𝕜 g⟫ =
      ∫ x, conj (f x) * g x ∂μ 
goal: ⟪(BoundedContinuousFunction.toLp 2 μ 𝕜) f, (BoundedContinuousFunction.toLp 2 μ 𝕜) g⟫_𝕜 =
  ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ

full_name: MeasureTheory.ContinuousMap.inner_toLp
code: /-- For continuous functions `f`, `g` on a compact, finite-measure topological space `α`, the L^2
inner product is the integral of their pointwise inner product. -/
theorem ContinuousMap.inner_toLp (f g : C(α, 𝕜)) :
    ⟪ContinuousMap.toLp (E := 𝕜) 2 μ 𝕜 f, ContinuousMap.toLp (E := 𝕜) 2 μ 𝕜 g⟫ =
      ∫ x, conj (f x) * g x ∂μ 
goal: ⟪(ContinuousMap.toLp 2 μ 𝕜) f, (ContinuousMap.toLp 2 μ 𝕜) g⟫_𝕜 = ∫ (x : α), (starRingEnd 𝕜) (f x) * g x ∂μ