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Let $X$ be a locally compact Hausdorff space, $C(X; K)$ the Banach space of continuous functions on $X$ with support in $K$, for compact $K \subseteq X$, and $C_c(X) = \lim_K C(X; K)$ the locally convex inductive limit with inclusion linking mappings $C(X; K) \to C_c(X)$. Denote the locally convex inductive limit topology on $C_c(X)$ by $\tau$, so that $\tau$ is the finest locally convex topology on $C_c(X)$ such that $C(X; K) \to C_c(X)$ is continuous. Denote also the norm topology by $\eta$. Then $\eta \subseteq \tau$. Moreover, $\tau$ is ultrabornological (therefore also barrelled and bornological), but $\eta$ is generally not barrelled.

Define $\eta^\beta$ as the strong topology $\beta(C_c, (C_c, \eta)')$ and $\eta^b$ as the associated barrelled topology (the coarsest barrelled topology finer than $\eta$; note that the finest locally convex topology is barrelled) and similarly $\eta^{ub}$ as the associated ultrabornological topology. Then

$$\eta \subseteq \eta^\beta \subseteq \eta^b \subseteq \eta^{ub} \subseteq \tau.$$

I would like to know whether $\eta^\beta = \tau$ or maybe just $\eta^b = \tau$ or at least $\eta^{ub} = \tau$. What if $X$ is assumed to be paracompact?

For $\sigma$-compact $X$ it holds $\eta^\beta = \tau$, see here (the proof uses a weighted seminorms approach).

Note also that for $X = [0, \omega_1)$ (the first uncountable ordinal), it holds $C_c(X) = C_0(X)$ (because continuous functions on $X$ are eventually constant) and therefore the norm topology $\eta$ is a Banach space topology. $X$ is not paracompact.

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    $\begingroup$ 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. $\endgroup$
    – user131781
    Commented Feb 25, 2020 at 16:01
  • $\begingroup$ @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 $\eta^\beta = \tau$). $\endgroup$
    – yada
    Commented Feb 27, 2020 at 15:06
  • $\begingroup$ 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 $A := \{ (a_x)_{x \in X} \in \varphi_d \mid \exists (b_x)_{x \in X} \in B, \, F \subseteq X \textrm{ finite}: a_x = b_x \textrm{ for } x \in F, a_x = 0 \textrm{ else} \}$. $\endgroup$
    – yada
    Commented Feb 27, 2020 at 16:58
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    $\begingroup$ 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. $\endgroup$
    – user131781
    Commented Feb 28, 2020 at 8:32

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