# point-wise approximation of the identity in hereditary Lindelof spaces

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).

• 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. – Taras Banakh May 29 '18 at 19:59