Theorems · Theorem · functional analysis
ContinuousLinearMap.opNorm_add_le
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : SeminormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : NontriviallyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [RingHomIsometric σ₁₂] (f g : E →SL[σ₁₂] F),
‖f + g‖ ≤ ‖f‖ + ‖g‖The operator norm satisfies the triangle inequality.
- Defined in
- Mathlib.Analysis.Normed.Operator.Basic
- Cited by
- 0 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.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Norm.normstatement and proof · cited by 5,413
- ContinuousLinearMapstatement and proof · cited by 5,352
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- add_mulproof · cited by 363
- LE.le.trans_eqproof · cited by 328
- RingHomIsometricstatement and proof · cited by 282
- add_nonnegproof · cited by 104
- ContinuousLinearMap.le_opNormproof · cited by 65
Cited by1
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.seminormproof · cited by 1