Theorems · Theorem · functional analysis
norm_eq_iInf_iff_real_inner_le_zero
∀ {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : InnerProductSpace ℝ F] {K : Set F},
Convex ℝ K → ∀ {u v : F}, v ∈ K → (‖u - v‖ = ⨅ w, ‖u - ↑w‖ ↔ ∀ w ∈ K, inner ℝ (u - v) (w - v) ≤ 0)Characterization of minimizers for the projection on a convex set in a real inner product space.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites49
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- Set.Elemstatement and proof · cited by 7,166
- Norm.normstatement and proof · cited by 5,413
- Set.rangeproof · cited by 4,705
- InnerProductSpacestatement and proof · cited by 3,523
- Nat.cast_oneproof · cited by 2,501
- le_antisymmproof · cited by 2,068
- Nat.cast_zeroproof · cited by 1,870
- iInfstatement and proof · cited by 1,690
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.norm_eq_iInf_iff_real_inner_eq_zeroproof · cited by 1