Theorems · Theorem · functional analysis

Bornology.IsVonNBounded.image_multilinear

∀ {ι : Type u_1} {𝕜 : Type u_2} {F : Type u_3} {E : ι → Type u_4} [inst : NormedField 𝕜]
  [inst_1 : (i : ι) → AddCommGroup (E i)] [inst_2 : (i : ι) → Module 𝕜 (E i)]
  [inst_3 : (i : ι) → TopologicalSpace (E i)] [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
  [inst_6 : TopologicalSpace F] [ContinuousSMul 𝕜 F] {s : Set ((i : ι) → E i)},
  Bornology.IsVonNBounded 𝕜 s → ∀ (f : ContinuousMultilinearMap 𝕜 E F), Bornology.IsVonNBounded 𝕜 (⇑f '' s)

The image of a von Neumann bounded set under a continuous multilinear map is von Neumann bounded. This version assumes that the codomain is a topological vector space.

Defined in

Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Cited by

0 results in Mathlib

Foundations

Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpaceContinuousSMul