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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?

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    $\begingroup$ 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 $\beta(X^*,X)$? $\endgroup$ – Francois Ziegler May 7 at 7:23
  • $\begingroup$ 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. $\endgroup$ – Jochen Wengenroth May 7 at 7:25
  • $\begingroup$ Good luck with the separable quotient problem! $\endgroup$ – Jochen Wengenroth May 7 at 7:26
  • $\begingroup$ 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. $\endgroup$ – Taras Banakh May 7 at 7:35
  • $\begingroup$ @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: $\sigma(X^*,X)=\beta^*_*(X^*,X)\subset\tau(X^*,X)\subset\beta(X^*,X)$. $\endgroup$ – Taras Banakh May 7 at 8:23

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