Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces.


**Question:** 
Why $BO$ is an $H$-space? My supervisor said "$BO=\Omega^\infty\mathbb{E}$ where $\mathbb{E}$ is a spectrum." What does this mean? 



classifying space of orthogonal groups
$BO$ is the connected component of the zeroth space of a spectrum called the [real K-theory spectrum](http://ncatlab.org/nlab/show/K-theory+spectrum). This spectrum represents a cohomology theory, namely real K-theory, and this means that $BO$ has much more structure than an H-space: it is in fact an [infinite loop space](http://ncatlab.org/nlab/show/infinite+loop+space), which is loosely a homotopy-theoretic version of an abelian group (as opposed to merely a monoid).


If you believe that homotopy classes of maps $X \to BO$ classify stable real vector bundles (ignoring their dimension) on $X$, then the H-space structure on $BO$ comes from direct sum of vector bundles. 


