Theorems · Theorem · convex and discrete geometry

ProperCone.dual_singleton

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommRing R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R]
  [inst_3 : TopologicalSpace R] [inst_4 : ClosedIciTopology R] [inst_5 : AddCommGroup M] [inst_6 : Module R M]
  [inst_7 : TopologicalSpace M] [inst_8 : AddCommGroup N] [inst_9 : Module R N] [inst_10 : TopologicalSpace N]
  {p : M →ₗ[R] N →ₗ[R] R} [inst_11 : p.IsContPerfPair] [inst_12 : IsTopologicalRing R] [inst_13 : OrderClosedTopology R]
  (x : M), ProperCone.dual p {x} = ProperCone.comap (p.toContPerfPair x) (ProperCone.positive R R)

The inner dual cone of a singleton is given by the preimage of the positive cone under the linear map p x.

Defined in

Mathlib.Analysis.Convex.Cone.Dual

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingPartialOrderIsOrderedRingTopologicalSpaceClosedIciTopologyAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpaceLinearMap.IsContPerfPairIsTopologicalRingOrderClosedTopology