# A new topology on the dual of a locally convex space?

Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if this topology had been considered in the literature.

It is defined as follows.

Let $$X$$ be a locally convex space over the field $$\mathbb F$$ of real or complex numbers, and $$X^*$$ be the linear space of all linear continuous functionals on $$X$$.

A subset $$F\subset X^*$$ is called weakly$$^*$$ bounded if for any $$x\in X$$ the set $$\{f(x):f\in F\}$$ is bounded in the field $$\mathbb F$$.

By $$\beta^*$$ we denote the family of weakly$$^*$$ bounded subsets of $$X^*$$.

A subset $$B\subset X$$ is called $$\beta^*$$-bounded if for any weakly$$^*$$ bounded subset $$F\subset X^*$$ the set $$\{f(x):f\in F,\;x\in B\}$$ is bounded in the field $$\mathbb F$$. The family of $$\beta^*$$-bounded subsets of $$X$$ is denoted by $$\beta^*_*$$.

It is easy to see that each $$\beta^*$$-bounded set $$B$$ in $$X$$ is bounded in the standard (von Neumann) sense: for any neighborhood $$U$$ of zero there exists $$n\in\mathbb N$$ with $$B\subset nU$$. So, the bornology $$\beta^*_*$$ is contained in the bornology $$\beta$$ of all bounded sets.

On the dual space $$X^*$$ consider the topology $$\beta^*_*(X^*,X)$$ of uniform convergence on $$\beta^*$$-bounded subsets of $$X$$.

It is clear that this topology contains the weak$$^*$$ topology $$\sigma(X^*,X)$$ of $$X^*$$ (i.e., the topology of uniform convergence on finite subsets of $$X$$) and is contained in the strong topology $$\beta(X^*,X)$$ (of uniform convergence on bounded subsets of $$X$$). So,

$$\sigma(X^*,X)\subset\beta^*_*(X^*,X)\subset\beta(X^*,X).$$

If $$X$$ is a Banach space, then $$\beta^*_*(X^*,X)=\beta(X^*,X)$$ is the topology generated by the dual norm on $$X^*$$.

But already for normed spaces the topologies $$\beta^*_*(X^*,X)$$ and $$\beta(X^*,X)$$ can be different. For example, if $$X=\ell^2_0$$ is the pre-Hilbert space of all eventally zero sequences in $$\ell_2$$, then $$\beta^*_*(X^*,X)=\sigma(X^*,X)\ne\beta(X^*,X)$$.

Problem. Has the topology $$\beta^*_*(X^*,X)$$ been considered in the literature? What about the bornology $$\beta^*_*$$ on $$X$$. Was it known?

• How does it relate to the Mackey topology $\tau(X^*,X)$ on $X^*$ (a.k.a. uniform convergence on $\sigma(X,X^*)$-compact convex subsets of $X$), which is also between $\sigma(X^*,X)$ and $\beta(X^*,X)$? – Francois Ziegler May 7 at 7:23
• The bornology $\beta_*^*(X,X^*)$ consists of all subsets of $X$ which are bounded with respect to the topology $\beta(X,X^*)=\beta(X,(X^*,\sigma(X^*,X))$ (uniform convergence on all weak$^*$-bounded subsets of $X^*$), right? (It might be appropriate to look at more general dual pair $\langle E,F\rangle$ to make the situation more symmetric.) The dual of $(X,\beta(X,X^*))$ can be strictly larger than $X^*$. Nevertheless, when $X^*$ is considered as a subspace of that dual, $\beta_*^*(X^*,X)$ is the relative topology of the strong topology. – Jochen Wengenroth May 7 at 7:25
• Good luck with the separable quotient problem! – Jochen Wengenroth May 7 at 7:26
• Saak Gabriyelyan pointed me out that the topology $\beta^*_*(X^*,X)$ can coincide with the topology $\beta^*(X^*,X)$ considered by Jarchow in his book on page 154. And the bornology $\beta^*_*$ is just the family of all bounded sets in the topology on $X$ generated by the neighborhood base at zero consisting of all barrels. – Taras Banakh May 7 at 7:35
• @FrancoisZiegler Surprisingly, but there is no definite inclusion relation between the topologies $\beta^*_*(X^*,X)$ and $\tau(X^*,X)$: For a barreled space $X$ we have $\tau(X^*,X)\subset \beta^*_*(X^*,X)=\beta(X^*,X)$, but for the pre-Hilbert space $X=\ell^2_0$ the inclusion is reversed: $\sigma(X^*,X)=\beta^*_*(X^*,X)\subset\tau(X^*,X)\subset\beta(X^*,X)$. – Taras Banakh May 7 at 8:23