Theorems · Theorem · functional analysis
norm_add_lt_of_not_sameRay
∀ {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, if x and y are not in the same ray, then ‖x + y‖ < ‖x‖ + ‖y‖.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
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.leproof · cited by 2,189
- LT.lt.ne'proof · cited by 1,417
- Metric.ballproof · cited by 735
- div_posproof · cited by 337
- SemigroupAction.mul_smulproof · cited by 291
- smul_addproof · cited by 263
- norm_smulproof · cited by 242
- div_selfproof · cited by 237
Cited by3
Results whose statement or proof uses this declaration.
- sameRay_iff_norm_addproof · cited by 3
- lt_norm_sub_of_not_sameRayproof · cited by 2
- Polynomial.cyclotomic_eval_lt_add_one_pow_totientproof · cited by 1