Theorems · Theorem · functional analysis
ContinuousLinearMap.sSup_sphere_eq_nnnorm
∀ {𝕜 : Type u_1} {𝕜₂ : Type u_2} {E : Type u_4} {F : Type u_5} [inst : NormedAddCommGroup E]
[inst_1 : SeminormedAddCommGroup F] [inst_2 : DenselyNormedField 𝕜] [inst_3 : NontriviallyNormedField 𝕜₂]
[inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜₂ F] {σ₁₂ : 𝕜 →+* 𝕜₂} [inst_6 : RingHomIsometric σ₁₂]
[NormedAlgebra ℝ 𝕜] (f : E →SL[σ₁₂] F), sSup ((fun x => ‖f x‖₊) '' Metric.sphere 0 1) = ‖f‖₊When the domain is a real normed space, ContinuousLinearMap.sSup_unitClosedBall_eq_nnnorm can
be tightened to take the supremum over only the Metric.sphere.
- Defined in
- Mathlib.Analysis.Normed.Operator.NNNorm
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites38
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- RingHomstatement and proof · cited by 10,189
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Set.imagestatement and proof · cited by 5,609
- Norm.normproof · cited by 5,413
- ContinuousLinearMapstatement and proof · cited by 5,352
- NNRealstatement and proof · cited by 4,310
- mul_oneproof · cited by 3,885
- SeminormedAddCommGroupstatement and proof · cited by 2,671
Cited by1
Results whose statement or proof uses this declaration.
- ContinuousLinearMap.sSup_sphere_eq_normproof · cited by 0