Let $E_0$ be a topological space. Are the following two definitions of an infinite loop space $E_0$ equivalent?
Definition 1: For each $n>0$, there exists a topological space $E_n$ such that $E_{n-1} = \Omega E_n$. So this is saying $E_0$ is the loop space of the loop space of the loop space of ... etc.
Definition 2: For each $n>0$, there exists a topological space $E_n$ such that $E_0 = \Omega^n E_n$.
It's clear that Definition 1 implies Definition 2. For the other way, based on Definition 2, it seems we can only deduce that for each $n>0$, $E_{n-1}$ is homotopic to $\Omega E_n$. So I guess Definition 1 is stronger than Definition 2?
Edit:
Every "$=$" above should be interpreted as "is homotopy equivalent to". I also used the phrase "is homotopic to" as a shortcut to "is homotopy equivalent to". I hope this won't cause confusions.