Timeline for Associated barrelled topology of norm topology on $C_c(X)$
Current License: CC BY-SA 4.0
5 events
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| Feb 28, 2020 at 8:32 | comment | added | user131781 | The reason I mentioned the discrete case is that it suggests the following path to a solution for the paracompact one. Consider next a direct sum of compacta where similar methods apply. Finally, reduce the general (paracompact) case to the latter by using the technique of partitions of unity (for inductive and projective limits of locally convex spaces—de Wilde). Since I haven’t sat down to write this up, I will leave it as a comment. I have no ideas about the non-paracompact case. | |
| Feb 27, 2020 at 16:58 | comment | added | yada | If I am not mistaken, for a discrete $X$ it also holds $\eta^\beta = \tau$ because $\beta(\varphi_d, \varphi_d) = \beta(\varphi_d, \omega_d)$: every bounded set $B \subseteq \omega_d$ is contained in the closure of a bounded set $A \subseteq \varphi_d$: set $A := \{ (a_x)_{x \in X} \in \varphi_d \mid \exists (b_x)_{x \in X} \in B, \, F \subseteq X \textrm{ finite}: a_x = b_x \textrm{ for } x \in F, a_x = 0 \textrm{ else} \}$. | |
| Feb 27, 2020 at 15:06 | comment | added | yada | @user131781 I must regret, that I am a novice to locally convex spaces, but I think that the question fits MO better than MSE. For countable discrete spaces (they are $\sigma$-compact), this follows from the link above: $\eta^\beta = \tau$. For a general discrete space $X$ with cardinality $d$ (so that $C_c = \varphi_d$ and $(C_c, \tau)' = \omega_d$, $\tau$ is the finest lc top.) one can find in [Köthe, "Topological Vector Spaces II", §34.10 Remark] that $\sigma(\varphi_d, \varphi_d)^b = \beta(\varphi_d, \omega_d) = \tau$, hence $\eta^b = \tau$. (I don't know whether $\eta^\beta = \tau$). | |
| Feb 25, 2020 at 16:01 | comment | added | user131781 | In a question like this it is often a good strategy to start by looking at the simplest non trivial case. Here this would be a discrete space, say the positive integers. Then the situation is quite transparent and might provide hints towards the general solution. | |
| Feb 25, 2020 at 8:22 | history | asked | yada | CC BY-SA 4.0 |