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Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively. A dual of $E^{\star}$ given by the $\beta(E^{\star\star}, E^{\star})$ topology is usually denoted by $E^{\star\star}$ and it is called a double dual of $E$.

Question 1

Is it true that the topology $\beta(E^{\star}, E^{\star\star})$ on $E^{\star}$ is always finer than $\beta(E^{\star}, E)$ (on $E^{\star}$), apart from the case when $E$ is reflexive and these topologies coincide.

Fact 1

It is well-known that if $E$ is an (F)-space then the (initial) (F)-space topology coincides with strong topology $\beta(E, E^{\star})$.

Question 2

Can we find a non-metrizable locally convex space for which the above sentence is true? (i.e. instead of (F)-space we put a non-metrizable locally convex space).

Thank you in advance for any help.

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1 Answer

up vote 4 down vote accepted

For the first question, the strong topology is the polar topology generated by all weakly bounded subsets. The weakly bounded subsets of $E$ are also weakly bounded in $E^{\ast\ast}$ since $E\subset E^{\ast\ast}$ and they have the same dual space $E^{\ast}.$ Therefore $\beta(E^{\ast},E^{\ast\ast})$ is finer than $\beta(E^{\ast},E).$

For the second question, did you try the space of test functions, i.e, infinitely differentiable functions with compact support? More generally, Hausdorff barrelled spaces have the property you want, so you should look for spaces in the class of barrelled spaces.

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I have found a theorem in "Topological vector spaces" of Adash, Ernst and Keim, that a lcs space $(E, \tau)$ is barelled if and only if $\tau=\beta(E, E^*)$. This answers the Question 2 completely in the class of lcs spaces. –  Tomek Kania Aug 17 '11 at 0:29
    
Thank you for the comment. In fact I don't know much barrelled spaces except the notions and some simple properties :-) –  Đức Anh Aug 17 '11 at 12:41
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