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For a dual pair $\langle X, Y \rangle$ of vector spaces $X$ and $Y$ denote by $\sigma(X,Y)$ the weak topology and by $\beta(X,Y)$ the strong topology on $X$. Consider the following known statement

(S) if $X$ is normed and $\sigma(X,X') = \beta(X,X')$ then $X$ is finite-dimensional.

There are infinite-dimensional locally convex spaces $X$ with $\sigma(X,X') = \beta(X,X')$. In fact, if $Y$ is any vector space and $Y^*$ the algebraic dual of $Y$ then $\sigma(Y^*, Y) = \beta(Y^*, Y)$. As an example, if $Y = \varphi$ (the space of sequences that are eventually $0$) with its largest locally convex topology then $Y' = Y^* = \omega$ (the space of all sequences) and thus $\sigma(\omega, \varphi) = \beta(\omega, \varphi)$. In other words, $X := \omega$ equipped with the topology $\sigma(\omega, \varphi)$ is an infinite-dimensional locally convex space for which its strong topology coincides with its weak topology. But note also that $X$ is a Fréchet space.

To which types of locally convex spaces $X$ can statement (S) be extended?

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For a (Hausdorff) locally convex space (l.c.s.) you have $\sigma(X,X')=\beta(X,X')$ if and only if all $\sigma(X',X)$-bounded subsets of $X'$ are finite dimensional (i.e., their linear span has finite dimension).

In general, this is a surprisingly rich class of l.c.s.: By an old result of Susanne Dierolf every l.c.s. is a quotient of such a space (which is also complete).

On the other hand, if $X$ is barrelled with $\sigma(X,X')=\beta(X,X')$ then the $\sigma(X',X)$-bounded sets are equicontinuous and this implies that that all continuous semi-norms of $X$ have co-finite dimensional kernel (so that you are very close to a product of lines).

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  • $\begingroup$ Thank you. Do you possible know of some more explicit classification of dual pairs $\langle X, Y \rangle$ satisfying $\sigma(X, Y) = \beta(X,Y)$, e.g. involving direct sums or products of real lines (or complex planes)? Or is it just hopeless to write down such an explicit list? $\endgroup$
    – yada
    Apr 11 '17 at 12:46

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