Theorems · Theorem · functional analysis

ContinuousMultilinearMap.opNorm_le_bound

∀ {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 : ℝ}, 0 ≤ M → (∀ (m : (i : ι) → E i), ‖f m‖ ≤ M * ∏ i, ‖m i‖) → ‖f‖ ≤ M

If one controls the norm of every f x, then one controls the norm of f.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

21 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype