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).

The file is a stub, some TODOs below.

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.

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 Banach-Alaoglu theorem (general version maybe in topology.algebra.module.weak_dual).
• Add metrizability of the dual unit ball (more generally bounded subsets) of weak_dual 𝕜 E under the assumption of separability of E. Sequential Banach-Alaoglu theorem would then follow from the general one.

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.

References #

• https://en.wikipedia.org/wiki/Weak_topology#Weak-*_topology

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).