# Is every dual space with strong topology locally convex?

Here is a question about topological vector spaces. Consider a t.v.s. $$X$$. Than on the continuous dual you can define the strong topology, given by the uniform convergence on the bounded setes. Let $$B = \{A \mid A \, \mathrm{bounded} \}$$. Than the topology on the dual is induced by the family of seminorms $$\big\{\Vert.\Vert_A\big\}_{A \in B}$$ $$\Vert f\Vert_A = \sup_{x \in A} \mid f(x)\mid$$

Now, we know that a topological vector space is locally convex if and only if its topology is induced by a family of seminorms.

Does this mean that every dual space with this topology is locally convex? Or I'm missing something?

• Exactly. The weak and weak* topologies are locally convex. Caution!: You defined the Strong Topology in $X'$ with the collection B={$A\subset X$; A bounded}, but the definition must be with the collection C={$A\subset X$; A is $\sigma(X,X')$-bounded} (this ensures the good definition of the seminorms). In special cases B coincides with C (in Locally Convex Hausdorff Spaces for example, see Narici and Beckenstein Thm 8.8.7.), but it is not true in general.