L^2 space #
If E is an inner product space over 𝕜 (ℝ or ℂ), then Lp E 2 μ
(defined in Mathlib/MeasureTheory/Function/LpSpace.lean)
is also an inner product space, with inner product defined as inner f g := ∫ a, ⟪f a, g a⟫ ∂μ.
Main results #
mem_L1_inner: forfandginLp E 2 μ, the pointwise inner productfun x ↦ ⟪f x, g x⟫belongs toLp 𝕜 1 μ.integrable_inner: forfandginLp E 2 μ, the pointwise inner productfun x ↦ ⟪f x, g x⟫is integrable.L2.innerProductSpace:Lp E 2 μis an inner product space.
theorem
MeasureTheory.MemLp.integrable_sq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : Measure α}
{f : α → ℝ}
(h : MemLp f 2 μ)
:
Integrable (fun (x : α) => f x ^ 2) μ
theorem
MeasureTheory.memLp_two_iff_integrable_sq_norm
{α : Type u_1}
{F : Type u_2}
{m : MeasurableSpace α}
{μ : Measure α}
[NormedAddCommGroup F]
{f : α → F}
(hf : AEStronglyMeasurable f μ)
:
theorem
MeasureTheory.memLp_two_iff_integrable_sq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : Measure α}
{f : α → ℝ}
(hf : AEStronglyMeasurable f μ)
:
theorem
MeasureTheory.MemLp.const_inner
{α : Type u_1}
{m : MeasurableSpace α}
{p : ENNReal}
{μ : Measure α}
{E : Type u_2}
{𝕜 : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(c : E)
{f : α → E}
(hf : MemLp f p μ)
:
theorem
MeasureTheory.MemLp.inner_const
{α : Type u_1}
{m : MeasurableSpace α}
{p : ENNReal}
{μ : Measure α}
{E : Type u_2}
{𝕜 : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{f : α → E}
(hf : MemLp f p μ)
(c : E)
:
theorem
MeasureTheory.Integrable.const_inner
{α : Type u_1}
{m : MeasurableSpace α}
{μ : Measure α}
{E : Type u_2}
{𝕜 : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{f : α → E}
(c : E)
(hf : Integrable f μ)
:
Integrable (fun (x : α) => inner 𝕜 c (f x)) μ
theorem
MeasureTheory.Integrable.inner_const
{α : Type u_1}
{m : MeasurableSpace α}
{μ : Measure α}
{E : Type u_2}
{𝕜 : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{f : α → E}
(hf : Integrable f μ)
(c : E)
:
Integrable (fun (x : α) => inner 𝕜 (f x) c) μ
theorem
integral_inner
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_2}
{𝕜 : Type u_3}
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
[CompleteSpace E]
[NormedSpace ℝ E]
{f : α → E}
(hf : MeasureTheory.Integrable f μ)
(c : E)
:
theorem
integral_eq_zero_of_forall_integral_inner_eq_zero
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_2}
(𝕜 : Type u_3)
[RCLike 𝕜]
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
[CompleteSpace E]
[NormedSpace ℝ E]
(f : α → E)
(hf : MeasureTheory.Integrable f μ)
(hf_int : ∀ (c : E), ∫ (x : α), inner 𝕜 c (f x) ∂μ = 0)
:
theorem
MeasureTheory.L2.eLpNorm_rpow_two_norm_lt_top
{α : Type u_1}
{F : Type u_3}
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup F]
(f : ↥(Lp F 2 μ))
:
theorem
MeasureTheory.L2.eLpNorm_inner_lt_top
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(f g : ↥(Lp E 2 μ))
:
instance
MeasureTheory.L2.instInnerSubtypeAEEqFunMemAddSubgroupLpOfNatENNReal
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
:
Equations
- MeasureTheory.L2.instInnerSubtypeAEEqFunMemAddSubgroupLpOfNatENNReal = { inner := fun (f g : ↥(MeasureTheory.Lp E 2 μ)) => ∫ (a : α), inner 𝕜 (↑↑f a) (↑↑g a) ∂μ }
theorem
MeasureTheory.L2.inner_def
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(f g : ↥(Lp E 2 μ))
:
@[deprecated _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.norm_sq_eq_re_inner (since := "2025-04-22")]
theorem
MeasureTheory.L2.norm_sq_eq_inner'
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(f : ↥(Lp E 2 μ))
:
Alias of _private.Mathlib.MeasureTheory.Function.L2Space.0.MeasureTheory.L2.norm_sq_eq_re_inner.
theorem
MeasureTheory.L2.mem_L1_inner
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(f g : ↥(Lp E 2 μ))
:
theorem
MeasureTheory.L2.integrable_inner
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
(f g : ↥(Lp E 2 μ))
:
Integrable (fun (x : α) => inner 𝕜 (↑↑f x) (↑↑g x)) μ
instance
MeasureTheory.L2.innerProductSpace
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
:
InnerProductSpace 𝕜 ↥(Lp E 2 μ)
Equations
- One or more equations did not get rendered due to their size.
theorem
MeasureTheory.L2.inner_indicatorConstLp_eq_setIntegral_inner
{α : Type u_1}
{E : Type u_2}
(𝕜 : Type u_4)
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{s : Set α}
(f : ↥(Lp E 2 μ))
(hs : MeasurableSet s)
(c : E)
(hμs : μ s ≠ ⊤)
:
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
MeasureTheory.L2.inner_indicatorConstLp_eq_inner_setIntegral
{α : Type u_1}
{E : Type u_2}
(𝕜 : Type u_4)
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{s : Set α}
[CompleteSpace E]
[NormedSpace ℝ E]
(hs : MeasurableSet s)
(hμs : μ s ≠ ⊤)
(c : E)
(f : ↥(Lp E 2 μ))
:
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
MeasureTheory.L2.inner_indicatorConstLp_one
{α : Type u_1}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
{s : Set α}
(hs : MeasurableSet s)
(hμs : μ s ≠ ⊤)
(f : ↥(Lp 𝕜 2 μ))
:
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
MeasureTheory.L2.inner_indicatorConstLp_indicatorConstLp
{α : Type u_1}
{E : Type u_2}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
[NormedAddCommGroup E]
[InnerProductSpace 𝕜 E]
{s t : Set α}
[CompleteSpace E]
(hs : MeasurableSet s)
(ht : MeasurableSet t)
(hμs : μ s ≠ ⊤ := by finiteness)
(hμt : μ t ≠ ⊤ := by finiteness)
(a b : E)
:
The inner product in L2 of two indicatorConstLps, i.e. functions which are constant a : E
and b : E on measurable s t : Set α with finite measure, respectively, is ⟪a, b⟫ times the
measure of s ∩ t.
theorem
MeasureTheory.L2.inner_indicatorConstLp_one_indicatorConstLp_one
{α : Type u_1}
{𝕜 : Type u_4}
[RCLike 𝕜]
[MeasurableSpace α]
{μ : Measure α}
{s t : Set α}
(hs : MeasurableSet s)
(ht : MeasurableSet t)
(hμs : μ s ≠ ⊤ := by finiteness)
(hμt : μ t ≠ ⊤ := by finiteness)
:
The inner product in L2 of indicators of two sets with finite measure
is the measure of the intersection.
theorem
MeasureTheory.L2.real_inner_indicatorConstLp_one_indicatorConstLp_one
{α : Type u_1}
[MeasurableSpace α]
{μ : Measure α}
{s t : Set α}
(hs : MeasurableSet s)
(ht : MeasurableSet t)
(hμs : μ s ≠ ⊤ := by finiteness)
(hμt : μ t ≠ ⊤ := by finiteness)
:
theorem
MeasureTheory.BoundedContinuousFunction.inner_toLp
{α : Type u_1}
{𝕜 : Type u_2}
[TopologicalSpace α]
[MeasurableSpace α]
[BorelSpace α]
[RCLike 𝕜]
(μ : Measure α)
[IsFiniteMeasure μ]
(f g : BoundedContinuousFunction α 𝕜)
:
inner 𝕜 ((BoundedContinuousFunction.toLp 2 μ 𝕜) f) ((BoundedContinuousFunction.toLp 2 μ 𝕜) g) = ∫ (x : α), g x * (starRingEnd 𝕜) (f x) ∂μ
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
MeasureTheory.ContinuousMap.inner_toLp
{α : Type u_1}
{𝕜 : Type u_2}
[TopologicalSpace α]
[MeasurableSpace α]
[BorelSpace α]
[RCLike 𝕜]
(μ : Measure α)
[IsFiniteMeasure μ]
[CompactSpace α]
(f g : C(α, 𝕜))
:
inner 𝕜 ((ContinuousMap.toLp 2 μ 𝕜) f) ((ContinuousMap.toLp 2 μ 𝕜) g) = ∫ (x : α), g x * (starRingEnd 𝕜) (f x) ∂μ
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.