**Question 1** Let $X$ a separable Banach Space and $Y\subset X$ linear subspace. When can we put a norm on $Y$ in such a way so that $Y$ is a Banach space?

Clearly if $Y$ is closed in the norm topology that's sufficient.

So, I'm coming into this with the perspective of Polish spaces(i.e. completely metrizable separable spaces) where a $G_\delta$ subset of a Polish space is once again Polish and every Polish space is a $G_\delta$ subset of a Polish space.

I was wondering if there is some sort of similar condition about complete norms and how the structure of the space matters.

I know there are linear spaces that can't be Banach spaces, e.g. $c_{0,0}$ the subspace of $c_0$ where cofinitely many elements in each sequence are $0$. In this case, there is a trivial Hamel basis, namely $e_i=(0,\ldots,0, 1, 0, \ldots)$ where $1$ is the $i$th element in the sequence. But this Hamel basis is countable and we know every Hamel basis of an infinite-dimensional Banach space must be uncountable.

We also know that the existence of a complete norm is equivalent to the existence of a complete homogeneous translation-invariant metric $d$, in the sense that

- $d$ is a complete metric
- $d(ax,ay)=|a|d(x,y)$
- $d(x+z,y+z)=d(x,y)$

From here, I believe we can see that $Y$ must at least have a complete metric and so must be a $G_\delta$ subset of $X$.

I haven't been able to come up with a counterexample of $Y$ a complete metric space with linear structure and $Y$ not a Banach Space, so I'm thinking that the $G_\delta$ may be a sufficient condition, so I think this can be restated as

**Question 2**
If $Y$ a linear space and $d$ a complete metric on $Y$. Then there exists a complete homogeneous translation-invariant metric $d'$ on $Y$.

Kind of related to the questions above, I know every vector space has a norm, just take a Hamel basis $(e_\lambda)$ and $x=\sum a_i e_i$. Now let $||x||=\max{|a_i|}$ or $||x||=\sum |a_i|$. Either of these forms a norm, but this norm isn't necessarily complete, especially in an infinite-dimensional space.