Sufficient conditions for the space of Radon measure to be a Banach space

Question

Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.

Usually, the additional assumptions on $\mathcal{X}$ are that it is locally compact. With this we have Riesz-Markov-Kakutani, i.e. the isometry between the dual of the continuous functions that vanish at infinity and the Radon measures. If $\mathcal{X}$ is no locally compact, a Stone-Cech compactification of $\mathcal{X}$ to $\beta\mathcal{X}$ can be used to establish duality between $C_0(\mathcal{X})$ and $M(\beta\mathcal{X})$.

I am interested in $M(\mathcal{X})$ when $\mathcal{X}$ is a Banach space. In my setting, Riesz-Markov-Kakutani is not relevant. Only that $M(\mathcal{X})$ is a Banach space.

Since $\mathcal{X}$ is no longer considered (locally) compact, an adapted definition of Radon measure may be needed. In the questions https://math.stackexchange.com/q/103208/653080 and https://mathoverflow.net/a/117693/137295, there are helpful lists of different definitions of Radon measures. However, these do not refer to $M(\mathcal{X})$ as a vector space. https://math.stackexchange.com/q/4485192/653080 suggests that $M(\mathcal{X})$ is a vector spaces if $\mathcal{X}$ is "sufficiently nice". I have been unable to find what "sufficiently nice" conditions could be, just that locally compact is one such condition.

Question: What are (minimal) sufficient additional conditions on $\mathcal{X}$ for $M(\mathcal{X})$ to be a Banach space?

Any pointers and references will be much appreciated.

Answer 1

If $X$ is a completely regular space, then $M(X)$ is identifiable with the dual of $C^b(X)$ when provided with the so-called strict topology and so is a Banach space. The latter topology was introduced by R.C. Buck in the 50‘s for locally compact $X$ using weighted seminorms, with the express purpose of extending the classical duality for compact $K$. His definition was then extended by several authors to the completely regular case, using various methods (replacing continuous weights by semi-continuous ones, using mixed topologies, the concepts of Saks spaces, $\dots$). The literature is rather comprehensive and you can easily find references by googling („bounded continuous functions“, „mixed topologies“, „Saks spaces“, “strict topologies“).