Theorems · Theorem · functional analysis

LinearMap.continuousAt_zero_of_locally_bounded

∀ {𝕜 : Type u_1} {𝕜' : Type u_2} {E : Type u_3} {F : Type u_4} [inst : AddCommGroup E] [inst_1 : TopologicalSpace E]
  [IsTopologicalAddGroup E] [inst_3 : AddCommGroup F] [inst_4 : TopologicalSpace F] [inst_5 : NontriviallyNormedField 𝕜]
  [inst_6 : Module 𝕜 E] [ContinuousSMul 𝕜 E] [inst_8 : NormedField 𝕜'] [inst_9 : Module 𝕜' F] {σ : 𝕜 →+* 𝕜'}
  [RingHomIsometric σ] [FirstCountableTopology E] (f : E →ₛₗ[σ] F),
  (∀ (s : Set E), Bornology.IsVonNBounded 𝕜 s → Bornology.IsVonNBounded 𝕜' (⇑f '' s)) → ContinuousAt (⇑f) 0

Defined in

Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Cited by

1 results in Mathlib

Foundations

Depth 126 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupTopologicalSpaceIsTopologicalAddGroupAddCommGroupTopologicalSpaceNontriviallyNormedFieldModuleContinuousSMulNormedFieldModuleRingHomIsometricFirstCountableTopology