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The space of finite signed Radon measures $M(\mathbb{R}^n)$ with total variation norm can be identified as either the dual space of continuous functions that vanishes at infinity $C_0(\mathbb{R}^n)$ or continuous function with bounded supported $C_c(\mathbb{R}^n)$ in the uniform norm. However, do they induce the same weak* topology on $M(\mathbb{R}^n)$?

Haoqing Yu
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    They coincide on bounded subsets of $M(\mathbb{R}^d)$. – Onur Oktay Mar 03 '24 at 08:28
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    The statement about locally convex spaces in this answer will probably interest you. – P. P. Tuong Mar 03 '24 at 08:34
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    As the answer linked in P. P. Tuong’s comment suggested, the two topologies are not the same since they give different dual spaces. However, because $C_c(\mathbb{R}^d)$ is uniformly dense in $C_0(\mathbb{R}^d)$, a standard $3\epsilon$-argument shows that the two topologies coincide on any bounded subset, as Onur Oktay’s comment mentioned. – David Gao Mar 03 '24 at 08:41
  • The restrictions of the two topologies to the sets $B_r = {\mu\in M(\mathbb{R}^d): |\mu|_{tv}\leq r }$ are the same. They do not coincide on $M(\mathbb{R}^d)$. – Onur Oktay Mar 03 '24 at 08:42
  • @DavidGao Thank you very much. I see it now. – Haoqing Yu Mar 03 '24 at 08:42
  • @OnurOktay Yes, sorry for my mistake. – Haoqing Yu Mar 03 '24 at 08:43
  • For example, take $(t_n)\subset \mathbb{R}$, $t_n\to\infty$. Let $\mu_n = n\delta_{t_n}$. Then $\mu_n(g) = ng(t_n)\to 0$ for each $g\in C_c(\mathbb{R})$, but not for some $g\in C_0(\mathbb{R})$. – Onur Oktay Mar 03 '24 at 08:45
  • @OnurOktay Great example. I was considering some conterexample of escaping to infinity but I didn't think of adding a factor of $n$ to make it work. – Haoqing Yu Mar 03 '24 at 08:48

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