File: Mathlib/Analysis/LocallyConvex/Polar.lean

full_name: LinearMap.polar
code: /-- The (absolute) polar of `s : Set E` is given by the set of all `y : F` such that `‖B x y‖ ≤ 1`
for all `x ∈ s`. -/
def polar (s : Set E) : Set F 
goal: Set F

full_name: LinearMap.polar_mem_iff
code: theorem polar_mem_iff (s : Set E) (y : F) : y ∈ B.polar s ↔ ∀ x ∈ s, ‖B x y‖ ≤ 1 
goal: y ∈ B.polar s ↔ ∀ x ∈ s, ‖(B x) y‖ ≤ 1

full_name: LinearMap.polar_mem
code: theorem polar_mem (s : Set E) (y : F) (hy : y ∈ B.polar s) : ∀ x ∈ s, ‖B x y‖ ≤ 1 
goal: x ∈ s → ‖(B x) y‖ ≤ 1

full_name: LinearMap.zero_mem_polar
code: @[simp]
theorem zero_mem_polar (s : Set E) : (0 : F) ∈ B.polar s 
goal: 0 ∈ B.polar s

full_name: LinearMap.polar_eq_iInter
code: theorem polar_eq_iInter {s : Set E} : B.polar s = ⋂ x ∈ s, { y : F | ‖B x y‖ ≤ 1 } 
goal: B.polar s = ⋂ x ∈ s, {y | ‖(B x) y‖ ≤ 1}

full_name: LinearMap.polar_gc
code: /-- The map `B.polar : Set E → Set F` forms an order-reversing Galois connection with
`B.flip.polar : Set F → Set E`. We use `OrderDual.toDual` and `OrderDual.ofDual` to express
that `polar` is order-reversing. -/
theorem polar_gc :
    GaloisConnection (OrderDual.toDual ∘ B.polar) (B.flip.polar ∘ OrderDual.ofDual) 
goal: GaloisConnection (⇑OrderDual.toDual ∘ B.polar) (B.flip.polar ∘ ⇑OrderDual.ofDual)

full_name: LinearMap.polar_iUnion
code: @[simp]
theorem polar_iUnion {ι} {s : ι → Set E} : B.polar (⋃ i, s i) = ⋂ i, B.polar (s i) 
goal: B.polar (⋃ i, s i) = ⋂ i, B.polar (s i)

full_name: LinearMap.polar_union
code: @[simp]
theorem polar_union {s t : Set E} : B.polar (s ∪ t) = B.polar s ∩ B.polar t 
goal: B.polar (s ∪ t) = B.polar s ∩ B.polar t

full_name: LinearMap.polar_antitone
code: theorem polar_antitone : Antitone (B.polar : Set E → Set F) 
goal: Antitone B.polar

full_name: LinearMap.polar_empty
code: @[simp]
theorem polar_empty : B.polar ∅ = Set.univ 
goal: B.polar ∅ = Set.univ

full_name: LinearMap.polar_zero
code: @[simp]
theorem polar_zero : B.polar ({0} : Set E) = Set.univ 
goal: B.polar {0} = Set.univ

full_name: LinearMap.subset_bipolar
code: theorem subset_bipolar (s : Set E) : s ⊆ B.flip.polar (B.polar s) 
goal: s ⊆ B.flip.polar (B.polar s)

full_name: LinearMap.tripolar_eq_polar
code: @[simp]
theorem tripolar_eq_polar (s : Set E) : B.polar (B.flip.polar (B.polar s)) = B.polar s 
goal: B.polar (B.flip.polar (B.polar s)) = B.polar s

full_name: LinearMap.polar_weak_closed
code: /-- The polar set is closed in the weak topology induced by `B.flip`. -/
theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
    (B.polar s) 
goal: IsClosed (B.polar s)

full_name: LinearMap.polar_univ
code: theorem polar_univ (h : SeparatingRight B) : B.polar Set.univ = {(0 : F)} 
goal: B.polar Set.univ = {0}

full_name: LinearMap.polar_subMulAction
code: theorem polar_subMulAction {S : Type*} [SetLike S E] [SMulMemClass S 𝕜 E] (m : S) :
    B.polar m = { y | ∀ x ∈ m, B x y = 0 } 
goal: B.polar ↑m = {y | ∀ x ∈ m, (B x) y = 0}

full_name: LinearMap.polarSubmodule
code: /-- The polar of a set closed under scalar multiplication as a submodule -/
def polarSubmodule {S : Type*} [SetLike S E] [SMulMemClass S 𝕜 E] (m : S) : Submodule 𝕜 F 
goal: Submodule 𝕜 F