Theorems · Theorem · functional analysis
sameRay_iff_norm_add
∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E},
SameRay ℝ x y ↔ ‖x + y‖ = ‖x‖ + ‖y‖In a strictly convex space, two vectors x, y are in the same ray if and only if the triangle
inequality for x and y becomes an equality.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Norm.normstatement and proof · cited by 5,413
- LT.lt.neproof · cited by 872
- SameRaystatement and proof · cited by 149
- StrictConvexSpacestatement and proof · cited by 57
- SameRay.norm_addproof · cited by 7
- norm_add_lt_of_not_sameRayproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- eq_of_norm_eq_of_norm_add_eqproof · cited by 5
- not_sameRay_iff_norm_add_ltproof · cited by 1
- Complex.norm_add_eq_iffproof · cited by 0