---
tag: RG52DZ1A8Y
kind: concept
name: Strong dual space
msc: [46A20]
specializes: [4A82S5FVRT]
sources:
  - url: https://en.wikipedia.org/wiki/Strong_dual_space
same_as:
  wikidata: Q96406607
  wikipedia: Strong dual space
---

## Summary

The strong dual space of a topological vector space $E$ is its [[continuous dual|4A82S5FVRT]] $E'$ with a certain topology.
It is the topology of uniform convergence on the [[bounded subsets|3BAJ8ZTVPX]] of $E$.
For a normed space this is just the norm topology of the dual norm.
The strong dual is the default topology on $E'$ in the theory of reflexivity.
A locally convex space $E$ is reflexive when the canonical map from $E$ to the strong dual of its strong dual is a topological isomorphism.

## Explanations

### L1 human

There are several ways to say that two continuous linear forms are close.
The weakest asks that they agree nearly at each single vector.
The strong one asks that they agree nearly, uniformly, on every bounded set of vectors.
For a normed space, being close on the unit ball is the same as being close in the dual norm, so the strong dual is the usual normed dual.