# 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: $(E'', \tau(E'', E'))'' = (E', \beta(E',E''))'$.
• ... so $(E'', \tau(E'', E'))'' = (E', \beta(E',E''))' \supseteq (E', \tau(E', E''))' = E''$ and the inclusion can be strict.
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.