TITLE: H-space structure on infinite projective spaces
QUESTION [15 upvotes]: Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does somebody know an explicit way to describe this structure in the cases $K({\mathbb Z}/2{\mathbb Z},1)={\mathbb R}P^\infty$ and $K({\mathbb Z},2)={\mathbb C}P^{\infty}$?

REPLY [2 votes]: John Baez has a nice expository page about this called  Classifying spaces made easy. About two thirds down the page he talks about multiplicative structure on $\mathbb{C}P^\infty$