Let $X$ be a compact Hausdorff space. In https://www.math.uni-bonn.de/people/scholze/Analytic.pdf (Ch. 3) Scholze considers the space of signed Radon measures on $X$ equipped with the filtered colimit (aka inductive limit) topology of the (in the weak$^*$-topology) compact absolutely convex subsets $\mathcal{M}(X)_{\leq c}$. Here $\mathcal{M}(X)_{\leq c}$ denotes the subset of measures with total variation norm less or equal than $c$ with $c>0$.
Then he states that the resulting topology is a locally convex vector topology. I was wondering if the subsets $\mathcal{M}(X)_{\leq c}$ form a neighborhood basis of the origin. If the answer is yes, then I do not see why the resulting topology is not the same as the one induced by the total variation norm. If the answer is no, then I do not see how to show that this topology is a locally convex vector topology.
Any clarification on this would be really appreciated.
I followed @Kevin Arlin's advice and posted the question on MO https://mathoverflow.net/questions/377896/is-the-filtered-colimit-topology-on-the-space-of-signed-radon-measures-linear-an. It already got answered.