File: Mathlib/Analysis/Normed/Module/WeakDual.lean

full_name: NormedSpace.Dual.toWeakDual
code: /-- For normed spaces `E`, there is a canonical map `Dual 𝕜 E → WeakDual 𝕜 E` (the "identity"
mapping). It is a linear equivalence. -/
def toWeakDual : Dual 𝕜 E ≃ₗ[𝕜] WeakDual 𝕜 E 
goal: NormedSpace.Dual 𝕜 E ≃ₗ[𝕜] WeakDual 𝕜 E

full_name: NormedSpace.Dual.coe_toWeakDual
code: @[simp]
theorem coe_toWeakDual (x' : Dual 𝕜 E) : toWeakDual x' = x' 
goal: NormedSpace.Dual.toWeakDual x' = x'

full_name: NormedSpace.Dual.toWeakDual_eq_iff
code: @[simp]
theorem toWeakDual_eq_iff (x' y' : Dual 𝕜 E) : toWeakDual x' = toWeakDual y' ↔ x' = y' 
goal: NormedSpace.Dual.toWeakDual x' = NormedSpace.Dual.toWeakDual y' ↔ x' = y'

full_name: NormedSpace.Dual.toWeakDual_continuous
code: theorem toWeakDual_continuous : Continuous fun x' : Dual 𝕜 E => toWeakDual x' 
goal: Continuous fun x' => NormedSpace.Dual.toWeakDual x'

full_name: NormedSpace.Dual.continuousLinearMapToWeakDual
code: /-- For a normed space `E`, according to `toWeakDual_continuous` the "identity mapping"
`Dual 𝕜 E → WeakDual 𝕜 E` is continuous. This definition implements it as a continuous linear
map. -/
def continuousLinearMapToWeakDual : Dual 𝕜 E →L[𝕜] WeakDual 𝕜 E 
goal: NormedSpace.Dual 𝕜 E →L[𝕜] WeakDual 𝕜 E

full_name: NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology
code: /-- The weak-star topology is coarser than the dual-norm topology. -/
theorem dual_norm_topology_le_weak_dual_topology :
    (UniformSpace.toTopologicalSpace : TopologicalSpace (Dual 𝕜 E)) ≤
      (WeakDual.instTopologicalSpace : TopologicalSpace (WeakDual 𝕜 E)) 
goal: UniformSpace.toTopologicalSpace ≤ WeakDual.instTopologicalSpace

full_name: WeakDual.toNormedDual
code: /-- For normed spaces `E`, there is a canonical map `WeakDual 𝕜 E → Dual 𝕜 E` (the "identity"
mapping). It is a linear equivalence. Here it is implemented as the inverse of the linear
equivalence `NormedSpace.Dual.toWeakDual` in the other direction. -/
def toNormedDual : WeakDual 𝕜 E ≃ₗ[𝕜] Dual 𝕜 E 
goal: WeakDual 𝕜 E ≃ₗ[𝕜] NormedSpace.Dual 𝕜 E

full_name: WeakDual.toNormedDual_apply
code: theorem toNormedDual_apply (x : WeakDual 𝕜 E) (y : E) : (toNormedDual x) y = x y 
goal: (WeakDual.toNormedDual x) y = x y

full_name: WeakDual.coe_toNormedDual
code: @[simp]
theorem coe_toNormedDual (x' : WeakDual 𝕜 E) : toNormedDual x' = x' 
goal: WeakDual.toNormedDual x' = x'

full_name: WeakDual.toNormedDual_eq_iff
code: @[simp]
theorem toNormedDual_eq_iff (x' y' : WeakDual 𝕜 E) : toNormedDual x' = toNormedDual y' ↔ x' = y' 
goal: WeakDual.toNormedDual x' = WeakDual.toNormedDual y' ↔ x' = y'

full_name: WeakDual.isClosed_closedBall
code: theorem isClosed_closedBall (x' : Dual 𝕜 E) (r : ℝ) : IsClosed (toNormedDual ⁻¹' closedBall x' r) 
goal: IsClosed (⇑WeakDual.toNormedDual ⁻¹' Metric.closedBall x' r)

full_name: WeakDual.polar
code: /-- The polar set `polar 𝕜 s` of `s : Set E` seen as a subset of the dual of `E` with the
weak-star topology is `WeakDual.polar 𝕜 s`. -/
def polar (s : Set E) : Set (WeakDual 𝕜 E) 
goal: Set (WeakDual 𝕜 E)

full_name: WeakDual.polar_def
code: theorem polar_def (s : Set E) : polar 𝕜 s = { f : WeakDual 𝕜 E | ∀ x ∈ s, ‖f x‖ ≤ 1 } 
goal: WeakDual.polar 𝕜 s = {f | ∀ x ∈ s, ‖f x‖ ≤ 1}

full_name: WeakDual.isClosed_polar
code: /-- The polar `polar 𝕜 s` of a set `s : E` is a closed subset when the weak star topology
is used. -/
theorem isClosed_polar (s : Set E) : IsClosed (polar 𝕜 s) 
goal: IsClosed (WeakDual.polar 𝕜 s)

full_name: WeakDual.isClosed_image_coe_of_bounded_of_closed
code: /-- While the coercion `↑ : WeakDual 𝕜 E → (E → 𝕜)` is not a closed map, it sends *bounded*
closed sets to closed sets. -/
theorem isClosed_image_coe_of_bounded_of_closed {s : Set (WeakDual 𝕜 E)}
    (hb : IsBounded (Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s) :
    IsClosed (((↑) : WeakDual 𝕜 E → E → 𝕜) '' s) 
goal: IsClosed (DFunLike.coe '' s)

full_name: WeakDual.isCompact_of_bounded_of_closed
code: theorem isCompact_of_bounded_of_closed [ProperSpace 𝕜] {s : Set (WeakDual 𝕜 E)}
    (hb : IsBounded (Dual.toWeakDual ⁻¹' s)) (hc : IsClosed s) : IsCompact s 
goal: IsCompact s

full_name: WeakDual.isClosed_image_polar_of_mem_nhds
code: /-- The image under `↑ : WeakDual 𝕜 E → (E → 𝕜)` of a polar `WeakDual.polar 𝕜 s` of a
neighborhood `s` of the origin is a closed set. -/
theorem isClosed_image_polar_of_mem_nhds {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsClosed (((↑) : WeakDual 𝕜 E → E → 𝕜) '' polar 𝕜 s) 
goal: IsClosed (DFunLike.coe '' WeakDual.polar 𝕜 s)

full_name: NormedSpace.Dual.isClosed_image_polar_of_mem_nhds
code: /-- The image under `↑ : NormedSpace.Dual 𝕜 E → (E → 𝕜)` of a polar `polar 𝕜 s` of a
neighborhood `s` of the origin is a closed set. -/
theorem _root_.NormedSpace.Dual.isClosed_image_polar_of_mem_nhds {s : Set E}
    (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsClosed (((↑) : Dual 𝕜 E → E → 𝕜) '' NormedSpace.polar 𝕜 s) 
goal: IsClosed (DFunLike.coe '' NormedSpace.polar 𝕜 s)

full_name: WeakDual.isCompact_polar
code: /-- The **Banach-Alaoglu theorem**: the polar set of a neighborhood `s` of the origin in a
normed space `E` is a compact subset of `WeakDual 𝕜 E`. -/
theorem isCompact_polar [ProperSpace 𝕜] {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
    IsCompact (polar 𝕜 s) 
goal: IsCompact (WeakDual.polar 𝕜 s)

full_name: WeakDual.isCompact_closedBall
code: /-- The **Banach-Alaoglu theorem**: closed balls of the dual of a normed space `E` are compact in
the weak-star topology. -/
theorem isCompact_closedBall [ProperSpace 𝕜] (x' : Dual 𝕜 E) (r : ℝ) :
    IsCompact (toNormedDual ⁻¹' closedBall x' r) 
goal: IsCompact (⇑WeakDual.toNormedDual ⁻¹' Metric.closedBall x' r)