dual — Mathlib

English

The simplicial property translates to closure of the underlying set equality for PointedCones.

Русский

Симплитическое свойство передаётся через замыкание базового множества для PointedCones.

LaTeX

$$K.closure = L ↔ closure (K:Set E) = L$$

Lean4

/-- The dual cone of a set `s` with respect to a perfect pairing `p` is the cone consisting of all
points `y` such that for all points `x ∈ s` we have `0 ≤ p x y`. -/
def dual (s : Set M) : ProperCone R N where
  toSubmodule := PointedCone.dual p s
  isClosed' := PointedCone.isClosed_dual fun _ ↦ p.continuous_of_isContPerfPair

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