Given vector space $E$ over (arbitrary) field. How do I check if is an inverse limit of some diagram $E_\lambda \to E_\gamma$?
I am not sure but it seems to me that for $\mathbb{k}^{\mathbb{N}}$ there is a representation as a inverse limit namely it's the limit of the system
$span(e_1,...e_n) \to span(e_1..., e_{n-1}), e_i \mapsto e_i, i < n, e_n \mapsto 0 $.
It goes without saying that finite dimensional vector space is the limit of finite dimensional vector spaces.