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.
