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I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone.

Theorem. The linear hull of any linearly independent Borel set in a Polish topological vector space $X$ is a Borel subset of $X$.

The proof easily follows from the Lusin-Souslin Theorem stating that injective continuous images of Borel sets are Borel.

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  • $\begingroup$ The ground field is an arbitrary Polish field? To what map do you apply the Lusin-Souslin theorem? $\endgroup$
    – YCor
    Nov 27, 2021 at 11:16
  • $\begingroup$ @YCor In fact, the proof is a bit tricky, but allows to get more: the linear hull of any linearly independent Lusin set in Lusin. A space is Lusin if it has a stronger Polish topology (which can be assumed to be zero-dimensional). Each Polish zero-dimensional space $P$ embeds into the real line and inherits the linear order from the real line. So, I apply the Lusin-Suslin Theorem to the injective map $(\mathbb R\setminus \{0\})^n\times \{(x_i)_{i\in n}\in P^n:x_0<x_1<...<x_{n-1}\}\to L$, $((\lambda_i),(x_i))\mapsto \sum_i\lambda_ix_i$. $\endgroup$
    – Taras Banakh
    Nov 27, 2021 at 11:37

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