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Let $X$ be a Hausdorff completely regular topological space, and let $C_b (X)$ be its algebra of continuous bounded functions. Endow $C_b (X)$ with a topology given by some seminorms, that contains the topology of pointwise convergence (I'm thinking of topologies like the topology of compact convergence or the strict topology, but I don't know how to better formulate this). Assume that there exists a basis $(b_i)_{i \in \mathbb N} \subset C_b (X)$, so that every $f \in C_b (X)$ can be written uniquely as $\sum _{i \in \mathbb N} f_i b_i$ with $f_i \in \mathbb C \ \forall i$.

  1. Is it possible to express the statement $f \in C_b (X)$ purely in terms of the coefficients $f_i$?
  2. Is it possible to express the statement $f_n \to 0$ purely in terms of the coefficients $f_{n,i}$ of each $f_n$?

In other words, I am trying to find out whether it is possible to "work in coordinates" in these topologies.

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    $\begingroup$ What makes you think that there will be a countable basis? Or do you mean a "basis" in some topological sense, like a Schauder basis? $\endgroup$
    – Igor Khavkine
    Commented Dec 2, 2017 at 14:13
  • $\begingroup$ Can such a basis exist? I don't think $C_b(X)$ is ever separable unless $X$ is compact (and second countable). $\endgroup$
    – Nik Weaver
    Commented Dec 2, 2017 at 15:40
  • $\begingroup$ @NikWeaver: I know that $C_b (X)$ is separable in the uniform topology iff $X$ is a compact metric space, but please notice that I use other topologies than the uniform one. $\endgroup$
    – Alex M.
    Commented Dec 2, 2017 at 18:54
  • $\begingroup$ @IgorKhavkine: A Schauder basis, of course. $\endgroup$
    – Alex M.
    Commented Dec 2, 2017 at 18:55

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