Is a closed set in a linear subspace of a normed space closed?

Question

Let $X$ be a normed space and let $Y$ be a linear subspace of $X$ with the norm given by the restriction of the norm on $X$ to $Y$. Let $S$ be a closed set in $Y$. So is $S$ closed in $X$?

What if $S\neq Y$? Can I get $S$ is closed in $X$?

Thank you in advance.

Answer 1

Not in general. Suppose that $Y$ itself is not a closed subset of $X$. Now, take $S=Y$.