# Equality of strong topologies for related dual pairs?

Let $$H$$ be a locally compact Hausdorff space and $$C_c(H)$$ the space of continuous functions with compact support. Equip $$C_c(H)$$ with the locally convex inductive limit topology $$\tau$$ of the Banach spaces $$C_K(H)$$ of continuous functions with support contained in a compact $$K \subseteq H$$, where $$K$$ runs over all compact subsets of $$H$$. Then $$(C_c(H), \tau)' = M(H)$$ is the space of real-valued Radon measures defined on the collection of relatively compact Borel sets. [$$C_c(H)$$ equipped with the (coarser) supremum norm $$\lVert \cdot \rVert$$ gives rise to the subspace $$(C_c(H), \lVert \cdot \rVert)' = M_b(H) \subseteq M(H)$$ of real-valued Radon measures defined on the whole Borel $$\sigma$$-algebra; the measures in $$M_b(H)$$ have bounded variation (in contrast to $$M(H)$$).]

Consider the dual pair $$\langle C_c(H), M(H) \rangle$$. For the strong topology it holds $$\beta(C_c(H), M(H)) = \tau\,$$ (because $$C_c(H)$$ is ultrabornological as an inductive limit of Banach spaces, hence barrelled). The linear hull $$D := \textrm{span}(\delta_x \mid x \in H)$$ of Dirac measures is (weakly-)dense in $$M(H)$$ and we can consider the dual pairs $$\langle C_c(H), D \rangle$$ and $$\langle C_c(H), M_b(H) \rangle$$. I want to know whether

$$\beta(C_c(H), D) = \beta(C_c(H), M(H)) \quad \textrm{or at least} \quad \beta(C_c(H), M_b(H)) = \beta(C_c(H), M(H)).$$

I have the following steps:

1. For a dual pair $$\langle Y, X \rangle$$ of vector spaces $$X, Y$$ and a (weakly-)dense subspace $$S \subseteq X$$ consider the dual pair $$\langle Y, S \rangle$$. For the strong topologies on $$Y$$ it then holds $$\beta(Y, S) \subseteq \beta(Y, X)$$. For the equality $$\beta(Y,S) = \beta(Y, X)$$ it is sufficient and necessary that every bounded set $$B \subseteq X$$ is contained in the (weak) closure of a bounded set $$A \subseteq S$$.

2. For simplicity let us first restrict to the case of a compact Hausdorff space $$K = H$$. Then $$\tau = \lVert \cdot \rVert$$ and $$C(K)' = M(K)$$. Equip $$M(K)$$ with the weak$$^*$$-topology. In order to apply 1., let $$B \subseteq M(H)$$ be a bounded set and show that there exists $$A \subseteq D$$ with $$B \subseteq \overline{A}$$. It holds $$B \subseteq r \overline{B}_1(0)$$ where $$r > 0$$ and $$\overline{B}_1(0) \subseteq M(H)$$ is the closed unit ball. It is enough to show that $$\overline{B}_1(0) \subseteq \overline{A}$$ for some bounded $$A \subseteq D$$. I know that the closed unit ball $$\overline{B}_1(0) \subseteq M(K)$$ is a weakly$$^*$$-compact convex set (Banach-Alaoglu). Also, the mapping $$K \to M(K)$$, $$x \mapsto \delta_x$$ is an embedding, so that we can identify $$K$$ as a compact subspace of $$M(K)$$. By Krein-Milman, $$\overline{B}_1(0)$$ is the closed convex hull of its extreme points, but the set of all the extreme points is the unit sphere $$\{ \mu \in M(H) \mid \lVert \mu \rVert = 1 \}$$ and this set is not contained in $$D$$. So, is it possible to find a suitable bounded set $$A \subseteq D$$? A canonical candidate for $$A$$ would be the absolutely convex hull $$\textrm{aco}(K)$$. If the strong topologies are not equal, what is then the dual of $$(C_c(H), \beta(C_c(H), D))$$ and $$(C_c(H), \beta(C_c(H), M_b(H)))$$?

3. If we can prove the conjecture for a compact space $$K$$, then I think it should also work for a more general locally compact space $$H$$, by relating $$C_c(H)$$ to its Banach space components $$C_K(H)$$ and considering polars of open neighborhoods in place of balls.

Note that for the special case $$H = \mathbb{N}$$ the conjecture is true: $$C_c(H) = \varphi$$ is the space of sequences that are eventually $$0$$, $$M(H) = C_c(H)' = \omega$$ is the space of all sequences, $$D = \textrm{span}(\mathbb{N}) = \varphi$$ and for this example, one can show that $$\beta(\varphi, \varphi) = \beta(\varphi, \omega)$$, because $$\beta(\varphi, \varphi)$$ is barrelled.

• Correction: it seems that the extreme points of $M(K)$ is indeed precisely the absolutely convex hull of Dirac measures. So the proof for the compact Hausdorff case should be fine.
I believe that your conjecture is true if $$H$$ is locally compact and $$\sigma$$-compact. Then the topology of the inductive limit $$C_c(K)=\lim\limits_\to C(K,H)$$ (where $$C(K,H)$$ denotes the Banach space of continuous functions with support in $$K$$) can be described by quite simple semi-norms of the form $$\|f\|_v=\sup\lbrace|f(x)|v(x): x\in H\rbrace$$ where $$v$$ is any strictly positive function on $$H$$ which is bounded on compact sets. It is clear that the restriction of $$\|\cdot\|_v$$ to each $$C(H,K)$$ is continuous there. It remains to show that these semi-norms indeed give the inductive limit topology. Fix a sequence $$K_n$$ of compact sets such that $$K_n$$ is contained in the interior of $$K_{n+1}$$ and $$H=\bigcup_n K_n$$. Then $$C_c(H)=\lim\limits_\to C(K_n,H)$$. Given any continuous semi-norm $$p$$ on $$C_c(H)$$ there are constants $$c_n>c_{n-1}>0$$ such that $$p(f)\le c_n \sup\lbrace|f(x)|:x\in H\rbrace$$ for all $$f\in C(K_n,X)$$. Next choose a continuous locally finite partition of unity $$\varphi_n$$ subordinated to the covering $$\mathring K_n$$ and set $$v=2^nc_n$$ on $$K_n\setminus K_{n-1}$$. Given any $$f\in C_c(H)$$ we can write $$f=\sum_{n\in \mathbb N} \varphi_nf$$ (where the sum is in fact finite) to obtain $$p(f)=\sum_{n\in \mathbb N} p(\varphi_nf) \le \sum_{n\in \mathbb N} 2^{-n} \|f\varphi_n\|_v\le \|f\|_v.$$
Now the proof should be as the one you suggested for the compact case: Given a $$0$$-neighbourhood $$U$$ of $$C_c(H)$$ it contains a unit ball with respect to some $$\|\cdot\|_v$$ and the extreme points of $$U^\circ$$ should be $$\pm v(x)\delta_x$$.
• Thank you. In the case of a $\sigma$-compact locally compact space, a reference to the literature for your result is [Summers, "Weighted Locally Convex Spaces of Continuous Functions", PhD thesis, (1968)] Theorem 3.36 (p. 60) together with Theorem 2.28 (p. 31). It remains open, what $\beta(C_c(H), D)$ and its dual (i.e. the "$C_c$-bidual" of $D$) actually is for a general locally compact space $H$.