# mathlibdocumentation

analysis.normed_space.weak_dual

# 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 normed_space.dual 𝕜 E or weak_dual 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 normed_space.dual 𝕜 E → weak_dual 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 → weak_dual 𝕜 E.

• normed_space.dual.to_weak_dual and weak_dual.to_normed_dual: Linear equivalences from dual 𝕜 E to weak_dual 𝕜 E and in the converse direction.
• normed_space.dual.continuous_linear_map_to_weak_dual: A continuous linear mapping from dual 𝕜 E to weak_dual 𝕜 E (same as normed_space.dual.to_weak_dual 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) weak_dual 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.
• weak_dual.is_compact_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 weak_dual _ E, if the nondiscrete normed field 𝕜 is proper as a topological space.
• weak_dual.is_compact_closed_ball (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 weak_dual 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.weak_dual.

When E is a normed space, the duals dual 𝕜 E and weak_dual 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 weak_dual.polar 𝕜 s for the "same" subset viewed as a subset of weak_dual 𝕜 E (a type synonym of the dual but with a different topology instance).

## 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 → weak_dual 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).

noncomputable def normed_space.dual.to_weak_dual {𝕜 : Type u_1} {E : Type u_2} [ E] :
≃ₗ[𝕜] E

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

Equations
@[simp]
theorem normed_space.dual.coe_to_weak_dual {𝕜 : Type u_1} {E : Type u_2} [ E] (x' : E) :
@[simp]
theorem normed_space.dual.to_weak_dual_eq_iff {𝕜 : Type u_1} {E : Type u_2} [ E] (x' y' : E) :
theorem normed_space.dual.to_weak_dual_continuous {𝕜 : Type u_1} {E : Type u_2} [ E] :
continuous (λ (x' : ,
noncomputable def normed_space.dual.continuous_linear_map_to_weak_dual {𝕜 : Type u_1} {E : Type u_2} [ E] :
→L[𝕜] E

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

Equations
theorem normed_space.dual.dual_norm_topology_le_weak_dual_topology {𝕜 : Type u_1} {E : Type u_2} [ E] :

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

noncomputable def weak_dual.to_normed_dual {𝕜 : Type u_1} {E : Type u_2} [ E] :
E ≃ₗ[𝕜]

For normed spaces E, there is a canonical map weak_dual 𝕜 E → dual 𝕜 E (the "identity" mapping). It is a linear equivalence. Here it is implemented as the inverse of the linear equivalence normed_space.dual.to_weak_dual in the other direction.

Equations
theorem weak_dual.to_normed_dual_apply {𝕜 : Type u_1} {E : Type u_2} [ E] (x : E) (y : E) :
= x y
@[simp]
theorem weak_dual.coe_to_normed_dual {𝕜 : Type u_1} {E : Type u_2} [ E] (x' : E) :
= x'
@[simp]
theorem weak_dual.to_normed_dual_eq_iff {𝕜 : Type u_1} {E : Type u_2} [ E] (x' y' : E) :
x' = y'
theorem weak_dual.is_closed_closed_ball {𝕜 : Type u_1} {E : Type u_2} [ E] (x' : E) (r : ) :

### Polar sets in the weak dual space #

def weak_dual.polar (𝕜 : Type u_1) {E : Type u_2} [ E] (s : set E) :
set 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 weak_dual.polar 𝕜 s.

Equations
theorem weak_dual.polar_def (𝕜 : Type u_1) {E : Type u_2} [ E] (s : set E) :
= {f : E | ∀ (x : E), x sf x 1}
theorem weak_dual.is_closed_polar (𝕜 : Type u_1) {E : Type u_2} [ E] (s : set E) :

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

theorem weak_dual.is_closed_image_coe_of_bounded_of_closed {𝕜 : Type u_1} {E : Type u_2} [ E] {s : set E)} (hc : is_closed s) :

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

theorem weak_dual.is_compact_of_bounded_of_closed {𝕜 : Type u_1} {E : Type u_2} [ E] [proper_space 𝕜] {s : set E)} (hc : is_closed s) :
theorem weak_dual.is_closed_image_polar_of_mem_nhds (𝕜 : Type u_1) {E : Type u_2} [ E] {s : set E} (s_nhd : s nhds 0) :

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

theorem normed_space.dual.is_closed_image_polar_of_mem_nhds (𝕜 : Type u_1) {E : Type u_2} [ E] {s : set E} (s_nhd : s nhds 0) :

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

theorem weak_dual.is_compact_polar (𝕜 : Type u_1) {E : Type u_2} [ E] [proper_space 𝕜] {s : set E} (s_nhd : s nhds 0) :

The Banach-Alaoglu theorem: the polar set of a neighborhood s of the origin in a normed space E is a compact subset of weak_dual 𝕜 E.

theorem weak_dual.is_compact_closed_ball (𝕜 : Type u_1) {E : Type u_2} [ E] [proper_space 𝕜] (x' : E) (r : ) :

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