H-space structure on infinite projective spaces 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 $K({\mathbb Z},2)={\mathbb C}P^{\infty}$?
 A: There's also a different way of writing down the $H$-space structure, that I like for its algebro-geometric flavor.  (I'll talk about $\mathbb{C}P^\infty$ here, and $\mathbb{R}P^\infty$ should be analogous.)
Regarding $\mathbb{C}P^\infty$ as a classifying space for complex line bundles, we know that this $H$-space structure is supposed to implement "tensor product of line bundles".  In a (not very explicit) sense this tells us the homotopy class of $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$: It represents the line bundle $\mathcal{O}(1,1) = p_1^* \mathcal{O}(1) \otimes p_2^* \mathcal{O}(1)$.  We can use this description to write down a much more explicit (and classical) explicit representative. 
First, let's recall what the analogous picture looks like for finite projective spaces.  The line bundle $\mathcal{O}(1,1)$ determines (upon picking generating sections) the Segre map
$\mathbb{C}P^n \times \mathbb{C}P^m \to \mathbb{C}P^{nm+n+m}$ which takes (in homogeneous coordinates) 
$([X_0:\ldots:X_n] , [Y_0:\ldots:Y_m]) \mapsto[X_0 Y_0: \ldots : X_i Y_j: \ldots: X_n Y_m]$ 
where I'm choosing to be vague on the precise ordering of the coordinates.
(In the end this won't matter up to homotopy, as the maps will become homotopic upon composing with $\mathbb{C}P^{nm+n+m} \hookrightarrow \mathbb{C}P^\infty$.)
The analogous formula with infinitely many homogeneous coordinate makes just as much sense, one just has to a good ordering of pairs of non-negative integers.  Such an infinite Segre map gives another realization of the $H$-space structure.
A: Look at $\mathbb R^\infty\setminus 0$ as the space of non-zero polynomials, which you can multiply. Pass to the quotient to construct the projective space and, from the multiplication, its $H$-space product.
The complex case is quite the same.
NB: Jason asks in a comment below if this is the same $H$-space structure that Hanno had in mind.
To check, we can use the fact that Hanno's is characterised by the fact that if $\mu:K(\mathbb Z_2,1)\times K(\mathbb Z_2,1)\to K(\mathbb Z_2,1)$ is his product and $\alpha\in H^1(K(\mathbb Z_2,1), \mathbb Z_2)$ is the class represented by the identify map $K(\mathbb Z_2,1)\to K(\mathbb Z_2,1)$, then $\mu^\*(\alpha)=\alpha\times 1+1\times\alpha$.
One should be able to check that this holds for the map given by multiplication of polynomials in a very small skeleton.
A: 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 $\mathbb{C}P^\infty$ 
