Infinite loop space

A space $X$ is an infinite loop space if there is a sequence $X_{0}=X,X_{1},\dots$ with homotopy equivalences $X_{n} \rightarrow \Omega X_{n+1}$, i.e. $X$ can be de-looped arbitrarily many times. By adjunction, we have maps $\Sigma X_{n} \rightarrow X_{n+1}$ so $\{X_{n}\}$ form a spectrum. Conversely there is a functor from spectra to infinite loop spaces given by sending a spectrum $\mathcal{E}$ to $\lim \Omega^{n} E_{n}$.