Zero eq top
QuantumMechanics.SpaceDHilbertSpace.SchwartzSubmodule.zero_eq_top
Project documentation
The linear equivalence between the Schwartz maps 𝓢(Space d, ℂ) and the Schwartz submodule of SpaceDHilbertSpace d μ. -/ def schwartzEquiv {d : ℕ} (μ : Measure (Space d)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] : 𝓢(Space d, ℂ) ≃ₗ[ℂ] SchwartzSubmodule d μ := LinearEquiv.ofInjective (schwartzIncl μ).toLinearMap (injective_toLp 2 μ) namespace Schwar...
Exact Lean statement
@[simp]
lemma zero_eq_top (μ : Measure (Space 0)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] :
SchwartzSubmodule 0 μ = ⊤Formal artifact
Lean source
@[simp]lemma zero_eq_top (μ : Measure (Space 0)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] : SchwartzSubmodule 0 μ = ⊤ := by ext ψ simp only [LinearMap.mem_range, ContinuousLinearMap.coe_coe, Submodule.mem_top, iff_true] let g : 𝓢(Space 0, ℂ) := { toFun x := ψ 0 smooth' := contDiff_const decay' k n := by refine ⟨‖ψ 0‖, fun x ↦ ?_⟩ rcases eq_zero_or_pos n with rfl | hn · rw [← one_mul ‖ψ 0‖] refine mul_le_mul ?_ (by simp) (norm_nonneg _) zero_le_one simp [Space.point_dim_zero_eq, zero_pow_le_one] · simp [iteratedFDeriv_const_of_ne hn.ne'] } use g ext filter_upwards [schwartzEquiv_coe_ae g] with x hg rw [← schwartzEquiv_apply_coe, hg, Space.point_dim_zero_eq x] rfl- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/QuantumMechanics/HilbertSpaces/SpaceD/SchwartzSubmodule.lean:100-120
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