ag.algebraic geometry - Infinite projective space

Starting with the affine case, if you try to define infinite dimensional affine space as Spec of k{x₁,x₂,...], then you realise that this is not a vector space of countable dimension, but something much larger. If you want a vector space over k of countable dimension, then this will not be a scheme, but instead will be an ind-scheme. A similar description should hold in the projective case.

Edit: Regarding why I am saying that Spec(k[x₁,x₂,...]) is too big: A (k-)point of Spec(k[x₁,x₂,...]) is an infinite sequence a₁,a₂,... of elements of k. If I wanted a vector space of countable dimension, then I should be asking for sequences a₁,a₂,... of elements of k, only finitely many of which are non-zero. This latter space is the inductive limit of affine n-space as n tends to infinity.