Theorems · Theorem · functional analysis

AlternatingMap.exists_bound_of_continuous

∀ {𝕜 : Type u} {E : Type wE} {F : Type wF} {ι : Type v} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : SeminormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : SeminormedAddCommGroup F]
  [inst_4 : NormedSpace 𝕜 F] [inst_5 : Fintype ι] (f : E [⋀^ι]→ₗ[𝕜] F),
  Continuous ⇑f → ∃ C, 0 < C ∧ ∀ (m : ι → E), ‖f m‖ ≤ C * ∏ i, ‖m i‖

If an alternating map in finitely many variables on a seminormed space 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.Alternating.Basic

Cited by

0 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintype