Weak dual of normed space

Let E be a normed space over a field 𝕜. This file is concerned with properties of the weak-* topology on the dual of E. By the dual, we mean either of the type synonyms StrongDual 𝕜 E or WeakDual 𝕜 E, depending on whether it is viewed as equipped with its usual operator norm topology or the weak-* topology.

It is shown that the canonical mapping StrongDual 𝕜 E → WeakDual 𝕜 E is continuous, and as a consequence the weak-* topology is coarser than the topology obtained from the operator norm (dual norm).

In this file, we also establish the Banach-Alaoglu theorem about the compactness of closed balls in the dual of E (as well as sets of somewhat more general form) with respect to the weak-* topology.

Main definitions

The main definitions concern the canonical mapping StrongDual 𝕜 E → WeakDual 𝕜 E.

• StrongDual.toWeakDual and WeakDual.toStrongDual: Linear equivalences from StrongDual 𝕜 E to WeakDual 𝕜 E and in the converse direction.
• NormedSpace.Dual.continuousLinearMapToWeakDual: A continuous linear mapping from StrongDual 𝕜 E to WeakDual 𝕜 E (same as StrongDual.toWeakDual but different bundled data).

Main results

The first main result concerns the comparison of the operator norm topology on StrongDual 𝕜 E and the weak-* topology on (its type synonym) WeakDual 𝕜 E:

• dual_norm_topology_le_weak_dual_topology: The weak-* topology on the dual of a normed space is coarser (not necessarily strictly) than the operator norm topology.
• WeakDual.isCompact_polar (a version of the Banach-Alaoglu theorem): The polar set of a neighborhood of the origin in a normed space E over 𝕜 is compact in WeakDual _ E, if the nontrivially normed field 𝕜 is proper as a topological space.
• WeakDual.isCompact_closedBall (the most common special case of the Banach-Alaoglu theorem): Closed balls in the dual of a normed space E over ℝ or ℂ are compact in the weak-star topology.

TODO

• Add that in finite dimensions, the weak-* topology and the dual norm topology coincide.
• Add that in infinite dimensions, the weak-* topology is strictly coarser than the dual norm topology.
• Add metrizability of the dual unit ball (more generally weak-star compact subsets) of WeakDual 𝕜 E under the assumption of separability of E.
• Add the sequential Banach-Alaoglu theorem: the dual unit ball of a separable normed space E is sequentially compact in the weak-star topology. This would follow from the metrizability above.

Implementation notes

Weak-* topology is defined generally in the file Mathlib/Topology/Algebra/Module/WeakDual.lean.

When M is a vector space, the duals StrongDual 𝕜 M and WeakDual 𝕜 M are type synonyms with different topology instances.

For the proof of Banach-Alaoglu theorem, the weak dual of E is embedded in the space of functions E → 𝕜 with the topology of pointwise convergence.

The polar set polar 𝕜 s of a subset s of E is originally defined as a subset of the dual StrongDual 𝕜 E. We care about properties of these w.r.t. weak-* topology, and for this purpose give the definition WeakDual.polar 𝕜 s for the "same" subset viewed as a subset of WeakDual 𝕜 E (a type synonym of the dual but with a different topology instance).

References

• https://en.wikipedia.org/wiki/Weak_topology#Weak-*_topology
• https://en.wikipedia.org/wiki/Banach%E2%80%93Alaoglu_theorem

Tags

weak-star, weak dual

def StrongDual.toWeakDual {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] : StrongDual R M ≃ₗ[R] WeakDual R M

For vector spaces M, there is a canonical map StrongDual R M → WeakDual R M (the "identity" mapping). It is a linear equivalence.

@[deprecated StrongDual.toWeakDual (since := "2025-08-3")]

def NormedSpace.Dual.toWeakDual {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] : StrongDual R M ≃ₗ[R] WeakDual R M

Alias of StrongDual.toWeakDual.

---

For vector spaces M, there is a canonical map StrongDual R M → WeakDual R M (the "identity" mapping). It is a linear equivalence.

@[simp]

theorem StrongDual.coe_toWeakDual {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] (x' : StrongDual R M) : toWeakDual x' = x'

@[deprecated StrongDual.coe_toWeakDual (since := "2025-08-3")]

theorem NormedSpace.Dual.coe_toWeakDual {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] (x' : StrongDual R M) : StrongDual.toWeakDual x' = x'

Alias of StrongDual.coe_toWeakDual.

@[simp]

theorem StrongDual.toWeakDual_inj {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] (x' y' : StrongDual R M) : toWeakDual x' = toWeakDual y' ↔ x' = y'

@[deprecated StrongDual.toWeakDual_inj (since := "2025-08-3")]

theorem NormedSpace.Dual.toWeakDual_inj {R : Type u_1} [CommSemiring R] [TopologicalSpace R] [ContinuousAdd R] [ContinuousConstSMul R R] {M : Type u_2} [AddCommMonoid M] [TopologicalSpace M] [Module R M] (x' y' : StrongDual R M) : StrongDual.toWeakDual x' = StrongDual.toWeakDual y' ↔ x' = y'

Alias of StrongDual.toWeakDual_inj.

def WeakDual.toStrongDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] : WeakDual 𝕜 E ≃ₗ[𝕜] StrongDual 𝕜 E

For vector spaces E, there is a canonical map WeakDual 𝕜 E → StrongDual 𝕜 E (the "identity" mapping). It is a linear equivalence. Here it is implemented as the inverse of the linear equivalence StrongDual.toWeakDual in the other direction.

theorem WeakDual.toStrongDual_apply {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (x : WeakDual 𝕜 E) (y : E) : (toStrongDual x) y = x y

@[simp]

theorem WeakDual.coe_toStrongDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (x' : WeakDual 𝕜 E) : toStrongDual x' = x'

@[simp]

theorem WeakDual.toStrongDual_inj {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (x' y' : WeakDual 𝕜 E) : toStrongDual x' = toStrongDual y' ↔ x' = y'

def WeakDual.polar (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (s : Set E) : Set (WeakDual 𝕜 E)

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.

theorem WeakDual.polar_def (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (s : Set E) : polar 𝕜 s = {f : WeakDual 𝕜 E | ∀ x ∈ s, ‖f x‖ ≤ 1}

theorem WeakDual.isClosed_polar (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [AddCommMonoid E] [TopologicalSpace E] [Module 𝕜 E] (s : Set E) : IsClosed (polar 𝕜 s)

The polar polar 𝕜 s of a set s : E is a closed subset when the weak star topology is used.

Weak star topology on duals of normed spaces

In this section, we prove properties about the weak-* topology on duals of normed spaces. We prove in particular that the canonical mapping StrongDual 𝕜 E → WeakDual 𝕜 E is continuous, i.e., that the weak-* topology is coarser (not necessarily strictly) than the topology given by the dual-norm (i.e. the operator-norm).

theorem NormedSpace.Dual.toWeakDual_continuous {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] : Continuous fun (x' : StrongDual 𝕜 E) => StrongDual.toWeakDual x'

def NormedSpace.Dual.continuousLinearMapToWeakDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] : StrongDual 𝕜 E →L[𝕜] WeakDual 𝕜 E

For a normed space E, according to toWeakDual_continuous the "identity mapping" StrongDual 𝕜 E → WeakDual 𝕜 E is continuous. This definition implements it as a continuous linear map.

theorem NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] : ContinuousLinearMap.uniformSpace.toTopologicalSpace ≤ instTopologicalSpaceWeakDual 𝕜 E

The weak-star topology is coarser than the dual-norm topology.

theorem WeakDual.isClosed_closedBall {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : StrongDual 𝕜 E) (r : ℝ) : IsClosed (⇑toStrongDual ⁻¹' Metric.closedBall x' r)

Polar sets in the weak dual space

theorem WeakDual.isClosed_image_coe_of_bounded_of_closed {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set (WeakDual 𝕜 E)} (hb : Bornology.IsBounded (⇑StrongDual.toWeakDual ⁻¹' s)) (hc : IsClosed s) : IsClosed (DFunLike.coe '' s)

While the coercion ↑ : WeakDual 𝕜 E → (E → 𝕜) is not a closed map, it sends bounded closed sets to closed sets.

theorem WeakDual.isCompact_of_bounded_of_closed {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] {s : Set (WeakDual 𝕜 E)} (hb : Bornology.IsBounded (⇑StrongDual.toWeakDual ⁻¹' s)) (hc : IsClosed s) : IsCompact s

theorem WeakDual.isClosed_image_polar_of_mem_nhds (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (s_nhds : s ∈ nhds 0) : IsClosed (DFunLike.coe '' polar 𝕜 s)

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 NormedSpace.Dual.isClosed_image_polar_of_mem_nhds (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} (s_nhds : s ∈ nhds 0) : IsClosed (DFunLike.coe '' StrongDual.polar 𝕜 s)

The image under ↑ : StrongDual 𝕜 E → (E → 𝕜) of a polar polar 𝕜 s of a neighborhood s of the origin is a closed set.

theorem WeakDual.isCompact_polar (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] {s : Set E} (s_nhds : s ∈ nhds 0) : IsCompact (polar 𝕜 s)

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 WeakDual.isCompact_closedBall (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] (x' : StrongDual 𝕜 E) (r : ℝ) : IsCompact (⇑toStrongDual ⁻¹' Metric.closedBall x' r)

The Banach-Alaoglu theorem: closed balls of the dual of a normed space E are compact in the weak-star topology.

theorem WeakDual.exists_countable_separating (𝕜 : Type u_3) (V : Type u_4) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [TopologicalSpace.SeparableSpace V] : ∃ (gs : ℕ → WeakDual 𝕜 V → 𝕜), (∀ (n : ℕ), Continuous (gs n)) ∧ ∀ ⦃x y : WeakDual 𝕜 V⦄, x ≠ y → ∃ (n : ℕ), gs n x ≠ gs n y

In a separable normed space, there exists a sequence of continuous functions that separates points of the weak dual.

theorem WeakDual.metrizable_of_isCompact (𝕜 : Type u_3) (V : Type u_4) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [TopologicalSpace.SeparableSpace V] (K : Set (WeakDual 𝕜 V)) (K_cpt : IsCompact K) : TopologicalSpace.MetrizableSpace ↑K

A compact subset of the dual space of a separable space is metrizable.

theorem WeakDual.isSeqCompact_of_isBounded_of_isClosed (𝕜 : Type u_3) (V : Type u_4) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [TopologicalSpace.SeparableSpace V] [ProperSpace 𝕜] {s : Set (WeakDual 𝕜 V)} (hb : Bornology.IsBounded (⇑StrongDual.toWeakDual ⁻¹' s)) (hc : IsClosed s) : IsSeqCompact s

theorem WeakDual.isSeqCompact_polar (𝕜 : Type u_3) (V : Type u_4) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [TopologicalSpace.SeparableSpace V] [ProperSpace 𝕜] {s : Set V} (s_nhd : s ∈ nhds 0) : IsSeqCompact (polar 𝕜 s)

The Sequential Banach-Alaoglu theorem: the polar set of a neighborhood s of the origin in a separable normed space V is a sequentially compact subset of WeakDual 𝕜 V.

theorem WeakDual.isSeqCompact_closedBall (𝕜 : Type u_3) (V : Type u_4) [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [TopologicalSpace.SeparableSpace V] [ProperSpace 𝕜] (x' : StrongDual 𝕜 V) (r : ℝ) : IsSeqCompact (⇑toStrongDual ⁻¹' Metric.closedBall x' r)

The Sequential Banach-Alaoglu theorem: closed balls of the dual of a separable normed space V are sequentially compact in the weak-* topology.