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 $X_n\to \Omega X_{n+1}$, i.e. $X$ can be de-looped arbitrarily many times. By adjunction, we have maps $\Sigma X_n\to X_{n+1}$ 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 $\lim {\Omega }^n{E}_n$.