Theorems · Theorem · functional analysis

MultilinearMap.exists_bound_of_continuous

∀ {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 : MultilinearMap 𝕜 E G),
  Continuous ⇑f → ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖

If a multilinear map in finitely many variables on normed spaces is continuous, then it satisfies the inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖, for some C which can be chosen to be positive.

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

2 results in Mathlib

Foundations

Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype