Theorems · Theorem · functional analysis

ContinuousLinearMap.sSup_sphere_eq_norm

∀ {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} [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_norm can be tightened to take the supremum over only the Metric.sphere.

Defined in

Mathlib.Analysis.Normed.Operator.NNNorm

Cited by

0 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupSeminormedAddCommGroupDenselyNormedFieldNontriviallyNormedFieldNormedSpaceNormedSpaceRingHomIsometricNormedAlgebra