Theorems · Theorem · functional analysis

ContinuousMultilinearMap.le_opNorm

∀ {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)]
  [inst_3 : SeminormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G] [inst_5 : Fintype ι]
  (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i), ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖

The fundamental property of the operator norm of a continuous multilinear map: ‖f m‖ is bounded by ‖f‖ times the product of the ‖m i‖.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

28 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype