Increasing sequence of strong topologies

Let $(X, \tau_0)$ be a locally convex Hausdorff space and define by induction:

$Y_1 := (X, \tau_0)'$, $Y_2 := (X, \beta(X,Y_1))'$, $Y_3 := (X, \beta(X,Y_2))'$, $\dots$

Here $\beta(X,Y)$ denotes the strong topology of the dual pair $\langle X, Y \rangle$. If $\tau(X,Y)$ denotes the Mackey topology then we get the following inclusions:

1. $Y_1 \subseteq Y_2 \subseteq Y_3 \subseteq \dots \subseteq X^*$ and
2. $\tau(X,Y_1) \subseteq \beta(X,Y_1) \subseteq \tau(X,Y_2) \subseteq \beta(X,Y_2) \subseteq \tau(X,Y_3) \subseteq \beta(X,Y_3) \subseteq \dots$

We can also extend to a transfinite induction by defining for a limit ordinal $\alpha$ the vector space $Y_\alpha$ as the inductive limit (= here union) of all the $Y_\beta$ for $\beta < \alpha$.

The questions at this point are:

1. Does the sequence $Y_n$ (or the "net" $Y_\beta$) stabilize? (Note that if the net $Y_\beta$ does not stabilize then it "converges" to $X^*$ in the sense that for every $f \in X^*$ there is $\beta$ with $f \in Y_\beta$.)
2. Do the mentioned topologies stabilize?

Regarding 1., I tried the following obvious example: Let $E$ be a Banach space and set $X := E'$ and $\tau_0 := \tau(E',E)$. Then $Y_1 = E$ and $Y_2 = E''$. Now it holds $Y_3 = (E', \beta(E', E''))' = E'' = Y_2$ if and only if $\beta(E', E'') = \tau(E', E'')$ and this is true if and only if $E''$ is reflexive (and thus $E$ is reflexive). So any non-reflexive Banach space $E$ provides an example for a space $(X, \tau_0)$ such that $Y_2 \subsetneq Y_3$. But does the sequence $Y_n$ stabilize later? (Is $Y_3 = E''''$ and so on?)

Regarding 1., as Daniel Fischer points out in the comments, for all spaces of the form $(X, \tau_0) = (E', \sigma)$ where $E$ is a space with barrelled strong dual $(E', \beta(E', E))$ and $\sigma$ any topology compatible with $\langle E', E'' \rangle$ it follows that $Y_2 = Y_3$, so the sequence stabilizes at the latest for $n = 2$. (Reason: Since $E'' = (E', \beta(E', E))'$ (simply by definition) it follows that $\beta(E', E) \subseteq \tau(E', E'')$ and if $\beta(E', E)$ is also barrelled then $\beta(E', E)$ is the strong topology $\beta(E', E'')$ of the dual pair $\langle E', E'' \rangle$.) However, there are spaces $E$ for which $\beta(E',E)$ is not barrelled, e.g. take any non-distinguished Fréchet space for $E$.

Regarding 2., the stabilization of the (strong) topologies corresponds to the stabilization of the spaces $Y_n$. In addition, [Komura, "Some Examples on Linear Topological Spaces" (1964)] has constructed a Fréchet space $E$ such that $\tau(E', E'')$ is bornological but $\beta(E', E)$ is not. In particular it follows that $\beta(E', E) \subsetneq \tau(E',E'')$. This provides an example for a space $(X, \tau_0)$ such that $\beta(X,Y_1) \subsetneq \tau(X,Y_2)$ (set $X := E'$ and $\tau_0 := \tau(E', E)$ and it follows $Y_1 = E$ and $Y_2 = E''$).

• Your Banach space example doesn't work. Since the norm topology on $E'$ (that is, $\beta(E',E)$) is barrelled, it follows that $\beta(E',E) = \tau(E',E'') = \beta(E',E'')$, so $Y_3 = Y_2$. May 29 '17 at 13:37
• @DanielFischer True. Thank you for the correction. For the Banach space example I applied [Wilansky, "Modern Methods in Topological Vector Spaces", Theorem 10-2-4] which says $\tau(E', E'') = \beta(E', E'')$ if and only if $(E'', \tau(E'', E'))'' = E''$ ($\tau(E'',E')$ is semireflexive). I was confused by shortened notation and my error was the following: if $E$ is any locally convex space then $(E'', \tau(E'', E'))'' = ((E'', \tau(E'', E'))')' = (E')' = E''$, i.e. $\tau(E'',E')$ is semireflexive for any $E$ - but this is not true. Correction: $(E'', \tau(E'', E'))'' = (E', \beta(E',E''))'$.
May 29 '17 at 14:22
• ... so $(E'', \tau(E'', E'))'' = (E', \beta(E',E''))' \supseteq (E', \tau(E', E''))' = E''$ and the inclusion can be strict.
May 29 '17 at 14:25

This question was investigated by Bella Tsirulnikov in "Barrelledness and dual strong sequences in locally convex space" (2004) and "Bounded sets and dual strong sequences in locally convex spaces" (2006).

If $\langle E, F \rangle$ is a dual pair then set $G_1 := F$ and $G_{n+1} := E[\beta(E, G_n)]'$ for $n \in \mathbb{N}$. Then $\beta_n := \beta(E, G_n)$ forms a sequence of increasing strong topologies on $E$ and $G_n \subseteq E^*$ a sequence of increasing subspaces of the algebraic dual $E^*$. This process need not terminate and one can consider the subspace $F_2 := \bigcup_n G_n$ with the dual pair $\langle E, F_2 \rangle$ and repeat the process. This can be iterated again and used to define a particular transfinite iteration of strong topologies which then terminates at the associated barrelled topology.