Let $I$ be an infinite set and {$E_i: i\in I$} be a family of vector spaces. For each $i\in I$, let $b_i \in E_i$ be a fixed non zero element.
For a finite subset $J$ of $I$, let $∏_{i \in J} E_i$ be the finite product of the $E_i$ and for $J \subset K$, define the connecting map ${\varphi}_{KJ}:∏_{i \in J} E_i \to ∏_{i \in K} E_i$ such that
${\varphi}_{KJ} ((x_i)_{i\in J})= (y_i)_{i \in K}$, where $y_i= x_i$ if $i \in J$ and $y_i = b_i$ if $i \in K-J$.
Then clearly, $(∏_{i \in J} E_i, {\varphi}_{KJ})_{ J \subset K \subset I}$ is a directed system of vector spaces therefore its direct limit exists. Can anybody tell me how the direct limit will look like?