Theorems · Theorem · functional analysis

IsModuleTopology.continuous_bilinear_of_pi_fintype

∀ {R : Type u_1} [inst : TopologicalSpace R] [inst_1 : CommSemiring R] {B : Type u_2} [inst_2 : AddCommMonoid B]
  [inst_3 : Module R B] [inst_4 : TopologicalSpace B] [IsModuleTopology R B] {C : Type u_3} [inst_6 : AddCommMonoid C]
  [inst_7 : Module R C] [inst_8 : TopologicalSpace C] [IsModuleTopology R C] (ι : Type u_4) [Finite ι]
  (bil : (ι → R) →ₗ[R] B →ₗ[R] C), Continuous fun ab => (bil ab.1) ab.2

If n is finite and B,C are R-modules with the module topology, then any bilinear map Rⁿ × B → C is automatically continuous. Note that whilst this result works for semirings, for rings this result is superseded by IsModuleTopology.continuous_bilinear_of_finite_left.

Defined in

Mathlib.Topology.Algebra.Module.ModuleTopology

Cited by

1 results in Mathlib

Foundations

Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceCommSemiringAddCommMonoidModuleTopologicalSpaceIsModuleTopologyAddCommMonoidModuleTopologicalSpaceIsModuleTopologyFinite