Let $X$, $Y$ be Polish topological vector spaces, i.e. vector spaces with a complete metric for which the addition and the scalar multiplication are continuous. Let $f\colon X\to Y$ be a continuous, linear, and surjective mapping.
Question 1. Is $X/\ker(f)$ a Polish space? (Here $\ker(f)$ denotes the kernel of the linear transformation $f$.)
Question 2. Is the restriction $f|_{X/\ker(f)}$ a Borel isomorphism, i.e. both $f|_{X/\ker(f)}$ and $f|_{X/\ker(f)}^{-1}$ are Borel measurable?