the unique class such that the following axioms are fulfilled: Normalization: The Whitney class of the tautological line bundle over the real projective space is nontrivial, i.e., . Rank: and for i above the rank of E, , that is, Whitney product formula: , that is, the Whitney class of a direct sum is the cup product of the summands' classes. Naturality: for any real vector bundle and map , where denotes the pullback vector bundle. The uniqueness of these classes is proved for example, in section 17.2 – 17.6 in Husemoller or section 8 in Milnor and Stasheff. There are several proofs of the existence, coming from various constructions, with several different flavours, their coherence is ensured by the unicity statement. Definition via infinite Grassmannians The infinite Grassmannians and vector bundles This section describes a construction using the notion of classifying space. For any vector space V, let denote the Grassmannian, the space of n-dimensional linear subspaces of V, and denote the infinite Grassmannian . Recall that it is equipped with the tautological bundle a rank n vector bundle that can be defined as the subbundle of the trivial bundle of fiber V whose fiber at a point is the subspace represented by Ẃ. Let , be a continuous map to the infinite Grassmannian. Then, up to isomorphism, the bundle induced by the map f on X depends only on the homotopy class of the map [f]. The pullback operation thus gives a morphism from the set of maps modulo homotopy equivalence, to the set of isomorphism classes of vector bundles of rank n over X. (The important fact in this construction is that if X is a paracompact space, this map is a bijection. This is the reason why we call infinite Grassmannians the classifying spaces of vector bundles.) Now, by the naturality axiom (4) above, . So it suffices in principle to know the values of for all j. However, the cohomology ring is free on specific generators arising from a standard cell decomposition, and it then turns out that these generators are in fact just given by . Thus, for any rank-n bundle, , where f is the appropriate classifying map. This in particular provides one proof of the existence of the Stiefel–Whitney classes. The case of line bundles We now restrict the above construction to line bundles, ie we consider the space, of line bundles