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 NormedSpace.Dual 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 NormedSpace.Dual 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 Dual 𝕜 E → WeakDual 𝕜 E.

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

Main results

The first main result concerns the comparison of the operator norm topology on dual 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.

TODOs:

• 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.

Notations

No new notation is introduced.

Implementation notes

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

When E is a normed space, the duals Dual 𝕜 E and WeakDual 𝕜 E 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 Dual 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

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 Dual 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).

def NormedSpace.Dual.toWeakDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] : NormedSpace.Dual 𝕜 E ≃ₗ[𝕜] WeakDual 𝕜 E

For normed spaces E, there is a canonical map Dual 𝕜 E → WeakDual 𝕜 E (the "identity" mapping). It is a linear equivalence.

Equations

• NormedSpace.Dual.toWeakDual = LinearEquiv.refl 𝕜 (E →L[𝕜] 𝕜)

@[simp]

theorem NormedSpace.Dual.coe_toWeakDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : NormedSpace.Dual 𝕜 E) : NormedSpace.Dual.toWeakDual x' = x'

@[simp]

theorem NormedSpace.Dual.toWeakDual_eq_iff {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : NormedSpace.Dual 𝕜 E) (y' : NormedSpace.Dual 𝕜 E) : NormedSpace.Dual.toWeakDual x' = NormedSpace.Dual.toWeakDual y' ↔ x' = y'

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

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

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.

Equations

• NormedSpace.Dual.continuousLinearMapToWeakDual = let __src := NormedSpace.Dual.toWeakDual; { toLinearMap := ↑__src, cont := ⋯ }

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] : UniformSpace.toTopologicalSpace ≤ WeakDual.instTopologicalSpace

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

def WeakDual.toNormedDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] : WeakDual 𝕜 E ≃ₗ[𝕜] NormedSpace.Dual 𝕜 E

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.

Equations

• WeakDual.toNormedDual = LinearEquiv.symm NormedSpace.Dual.toWeakDual

theorem WeakDual.toNormedDual_apply {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : WeakDual 𝕜 E) (y : E) : (WeakDual.toNormedDual x) y = x y

@[simp]

theorem WeakDual.coe_toNormedDual {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : WeakDual 𝕜 E) : WeakDual.toNormedDual x' = x'

@[simp]

theorem WeakDual.toNormedDual_eq_iff {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x' : WeakDual 𝕜 E) (y' : WeakDual 𝕜 E) : WeakDual.toNormedDual x' = WeakDual.toNormedDual y' ↔ x' = y'

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

Polar sets in the weak dual space

def WeakDual.polar (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 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.

Equations

• WeakDual.polar 𝕜 s = ⇑WeakDual.toNormedDual ⁻¹' NormedSpace.polar 𝕜 s

theorem WeakDual.polar_def (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E) : WeakDual.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} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (s : Set E) : IsClosed (WeakDual.polar 𝕜 s)

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

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 (⇑NormedSpace.Dual.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 (⇑NormedSpace.Dual.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_nhd : s ∈ nhds 0) : IsClosed (DFunLike.coe '' WeakDual.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_nhd : s ∈ nhds 0) : IsClosed (DFunLike.coe '' NormedSpace.polar 𝕜 s)

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 WeakDual.isCompact_polar (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [ProperSpace 𝕜] {s : Set E} (s_nhd : s ∈ nhds 0) : IsCompact (WeakDual.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' : NormedSpace.Dual 𝕜 E) (r : ℝ) : IsCompact (⇑WeakDual.toNormedDual ⁻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.