Norm smooth Fun To Lp eq
DeGiorgi.norm_smoothFunToLp_eq
Plain-language statement
Norm of the L² class carried by a smooth test function.
Exact Lean statement
theorem norm_smoothFunToLp_eq
{Ω : Set E} (hΩ : IsOpen Ω) {u : E → ℝ}
(hu : IsSmoothTestOn Ω u) :
‖smoothFunToLp hΩ hu‖ =
(∫ x, ‖u x‖ ^ (2 : ℝ) ∂(volume.restrict Ω)) ^ (1 / (2 : ℝ))Formal artifact
Lean source
theorem norm_smoothFunToLp_eq {Ω : Set E} (hΩ : IsOpen Ω) {u : E → ℝ} (hu : IsSmoothTestOn Ω u) : ‖smoothFunToLp hΩ hu‖ = (∫ x, ‖u x‖ ^ (2 : ℝ) ∂(volume.restrict Ω)) ^ (1 / (2 : ℝ)) := by let _ := (inferInstance : NeZero d) rw [smoothFunToLp, Lp.norm_toLp] rw [MeasureTheory.toReal_eLpNorm (smoothTestWitness hΩ hu).memLp.aestronglyMeasurable] simpa using (MeasureTheory.lpNorm_eq_integral_norm_rpow_toReal (μ := volume.restrict Ω) (f := u) (p := (2 : ENNReal)) (by norm_num) (by simp) (smoothTestWitness hΩ hu).memLp.aestronglyMeasurable)- Project
- DeGiorgi
- License
- Apache-2.0
- Commit
- 4c1b3077d378
- Source
- DeGiorgi/WeakFormulation/SmoothTests.lean:193-204
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Related declarations
Ae eq of tendsto e Lp Norm sub
BareFunction.ae_eq_of_tendsto_eLpNorm_sub
Plain-language statement
Lp limit uniqueness: if f_n → g₁ and f_n → g₂ in eLpNorm, then g₁ =ᵐ g₂.
Source project: DeGiorgi
Person-level attribution pending.
E Lp Norm pi le sum component
BareFunction.eLpNorm_pi_le_sum_component
Plain-language statement
Vector eLpNorm ≤ sum of component eLpNorms for Pi-valued functions. Uses eLpNorm_mono_real for the pointwise bound together with eLpNorm_sum_le for ℝ-valued functions, avoiding Pi instance synthesis.
Source project: DeGiorgi
Person-level attribution pending.
Mem Lp of tendsto e Lp Norm
BareFunction.memLp_of_tendsto_eLpNorm
Plain-language statement
If f n → g in eLpNorm and each f n ∈ Lp, then g ∈ Lp, provided g is AEStronglyMeasurable. Avoids the Lp type entirely. The key observation: eLpNorm (f n - g) → 0 means eLpNorm (f N - g) < 1 for some N. Then eLpNorm g ≤ eLpNorm (f N - g) + eLpNorm (f N) < ∞.
Source project: DeGiorgi
Person-level attribution pending.