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import Mathlib.MeasureTheory.Integral.MeanInequalities | ||
import Mathlib.MeasureTheory.Function.L1Space | ||
import Mathlib.Analysis.NormedSpace.Dual | ||
import Mathlib.Analysis.NormedSpace.LinearIsometry | ||
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/-! We show that the dual space of `L^p` for `1 ≤ p < ∞`. | ||
See [Stein-Shakarchi, Functional Analysis, section 1.4] -/ | ||
noncomputable section | ||
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open Real NNReal ENNReal NormedSpace MeasureTheory | ||
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variable {α 𝕜 E E₁ E₂ E₃ : Type*} {m : MeasurableSpace α} {p q : ℝ≥0∞} | ||
{μ : Measure α} [NontriviallyNormedField 𝕜] | ||
[NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] | ||
[NormedAddCommGroup E₁] [NormedSpace 𝕜 E₁] [FiniteDimensional 𝕜 E₁] | ||
[NormedAddCommGroup E₂] [NormedSpace 𝕜 E₂] [FiniteDimensional 𝕜 E₂] | ||
[NormedAddCommGroup E₃] [NormedSpace 𝕜 E₃] [FiniteDimensional 𝕜 E₃] | ||
[MeasurableSpace E] [BorelSpace E] | ||
[MeasurableSpace E₁] [BorelSpace E₁] | ||
[MeasurableSpace E₂] [BorelSpace E₂] | ||
[MeasurableSpace E₃] [BorelSpace E₃] | ||
(L : E₁ →L[𝕜] E₂ →L[𝕜] E₃) | ||
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namespace ENNReal | ||
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/-- Two numbers `p, q : ℝ≥0∞` are conjugate if `p⁻¹ + q⁻¹ = 1`. | ||
This does allow for the case where one of them is `∞` and the other one is `1`, | ||
in contrast to `NNReal.IsConjExponent`. -/ | ||
structure IsConjExponent (p q : ℝ≥0∞) : Prop where | ||
inv_add_inv_conj : p⁻¹ + q⁻¹ = 1 | ||
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namespace IsConjExponent | ||
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lemma one_le_left (hpq : p.IsConjExponent q) : 1 ≤ p := sorry | ||
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lemma symm (hpq : p.IsConjExponent q) : q.IsConjExponent p := sorry | ||
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lemma one_le_right (hpq : p.IsConjExponent q) : 1 ≤ q := hpq.symm.one_le_left | ||
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/- maybe useful: formulate an induction principle. To show something when `p.IsConjExponent q` then it's sufficient to show it in the following cases: | ||
* you have `p q : ℝ≥0` with `p.IsConjExponent q` | ||
* `p = 1` and `q = ∞` | ||
* `p = ∞` and `q = 1` -/ | ||
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/- add various other needed lemmas below (maybe look at `NNReal.IsConjExponent` for guidance) -/ | ||
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/- Versions of Hölder's inequality. | ||
Note that the hard case already exists as `ENNReal.lintegral_mul_le_Lp_mul_Lq`. -/ | ||
#check ENNReal.lintegral_mul_le_Lp_mul_Lq | ||
#check ContinuousLinearMap.le_opNorm | ||
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theorem lintegral_mul_le (μ : Measure α) {f : α → E₁} {g : α → E₂} | ||
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : | ||
∫⁻ a, ‖L (f a) (g a)‖₊ ∂μ ≤ ‖L‖₊ * snorm f p μ * snorm g q μ := by | ||
sorry | ||
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theorem integrable_bilin (μ : Measure α) {f : α → E₁} {g : α → E₂} | ||
(hf : Memℒp f p μ) (hg : Memℒp g q μ) : | ||
Integrable (fun a ↦ L (f a) (g a)) μ := by sorry | ||
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end IsConjExponent | ||
end ENNReal | ||
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namespace MeasureTheory | ||
namespace Lp | ||
-- note: we may need to restrict to `𝕜 = ℝ` | ||
variable | ||
[hpq : Fact (p.IsConjExponent q)] [h'p : Fact (p < ∞)] | ||
[hp : Fact (1 ≤ p)] [hq : Fact (1 ≤ q)] -- note: these are superfluous, but it's tricky to make them instances. | ||
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/- The map sending `g` to `f ↦ ∫ x, L (f x) (g x) ∂μ` induces a map on `L^p` into | ||
`Lp E₂ p μ →L[𝕜] E₃`. Generally we will take `E₃ = 𝕜`. -/ | ||
variable (p μ) in | ||
def toDual (g : Lp E₁ q μ) : Lp E₂ p μ →L[𝕜] E₃ := | ||
sorry | ||
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/- The map sending `g` to `f ↦ ∫ x, L (f x) (g x) ∂μ` is a linear isometry. -/ | ||
variable (p q μ) in | ||
def toDualₗᵢ : Lp E₁ q μ →ₗᵢ[𝕜] Lp E₂ p μ →L[𝕜] E₃ := | ||
sorry | ||
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/- The map sending `g` to `f ↦ ∫ x, L (f x) (g x) ∂μ` is a linear isometric equivalence. -/ | ||
variable (p q μ) in | ||
def dualIsometry (L : E₁ →L[𝕜] Dual 𝕜 E₂) : | ||
Dual 𝕜 (Lp E₂ p μ) ≃ₗᵢ[𝕜] Lp E q μ := | ||
sorry | ||
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end Lp | ||
end MeasureTheory |