Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$.

Q. Can we concluded that $X$ is hereditery Lindelof?

By an interesting example Taras Banakh proved that: $X$ is not necessarily written by countable union of second countable subsets (see here).

  • $\begingroup$ I think that the free (locally convex) linear topological space over the Sorgenfrey line will be a counterexample and encourage Ali Bagheri to try to prove this fact himself. $\endgroup$ – Taras Banakh May 29 '18 at 19:59

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