Let $X$ be any set. Prove there exist a vector space $V$ such that $X$ is a basis for $V$.
For example what would be the Vector space $V$ such that $X=\mathbb{N}$ is a basis? I don't really see any hint, perhaps if it's finite dimentional. Also we must take care about over which Field $V$ is a vector space.
For any countable set $X$ with members $x_i$ we can define the formal sums $\sum \lambda_i x_i$ where the coefficients $\lambda_i$ are members of a field such as $\mathbb{Q}$, $\mathbb{R}$ or $\mathbb{C}$. The set of formal sums becomes a vector space if we define addition and scalar multiplication in the natural way:
$(\sum \lambda_i x_i) + (\sum \kappa_i x_i) = \sum (\lambda_i + \kappa_i)x_i$
$\kappa (\sum \lambda_i x_i) = \sum (\kappa \lambda_i) x_i$