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refactor(Analysis): golf Mathlib/Analysis/Normed/Module/Dual #38493

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refactor(Analysis): golf Mathlib/Analysis/Normed/Module/Dual #38493

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refactor(Analysis): golf `Mathlib/Analysis/Normed/Module/Dual`
yuanyi-350 Apr 25, 2026
a4e7598
Update Mathlib/Analysis/Normed/Module/Dual.lean
yuanyi-350 Apr 25, 2026
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16 changes: 2 additions & 14 deletions Mathlib/Analysis/Normed/Module/Dual.lean
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Expand Up @@ -117,20 +117,8 @@ theorem polar_closedBall {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [

theorem polar_ball {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
(hr : 0 < r) : StrongDual.polar 𝕜 (ball (0 : E) r) = closedBall (0 : StrongDual 𝕜 E) r⁻¹ := by
apply le_antisymm
· intro x hx
rw [mem_closedBall_zero_iff]
apply le_of_forall_gt_imp_ge_of_dense
intro a ha
rw [← mem_closedBall_zero_iff, ← (mul_div_cancel_left₀ a (Ne.symm (ne_of_lt hr)))]
rw [← RCLike.norm_of_nonneg (K := 𝕜) (le_trans zero_le_one
(le_of_lt ((inv_lt_iff_one_lt_mul₀' hr).mp ha)))]
apply polar_ball_subset_closedBall_div _ hr hx
rw [RCLike.norm_of_nonneg (K := 𝕜) (le_trans zero_le_one
(le_of_lt ((inv_lt_iff_one_lt_mul₀' hr).mp ha)))]
exact (inv_lt_iff_one_lt_mul₀' hr).mp ha
· rw [← polar_closedBall hr]
exact LinearMap.polar_antitone _ ball_subset_closedBall
letI : NormedSpace ℝ E := .restrictScalars ℝ 𝕜 E
rw [← polar_closedBall hr, ← closure_ball _ hr.ne', polar_closure]

/-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms of all elements of
the polar `polar 𝕜 s` are bounded by a constant. -/
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