The splitting principle is as follows.



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> Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^\*: K^\*(X) \to K^\*(F(E))$ is injective and $p^\*(E)$ splits as the sum of line bundles.
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My question is, what is the idea/intuition behind the proof of the splitting principle?



Intuition/idea behind a proof of the splitting principle?
In the topological language you are using, $F(E)$ is the space of "orthogonal splittings". That is to say, $p^{-1}(x)$ is the space of all ways to write the fiber $E\_x$ as an orthogonal sum of one dimensional spaces. Since it is the "space of splittings", there is a tautological splitting over it. "$\square$"




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**Remark on alternative versions you may have seen:** It is more common to describe $F(E)$ as the space of flags. A (complete) flag $F\_{\bullet}$ in a vector space $V$ is a chain of subspaces $F\_1 \subset F\_2 \subset \cdots \subset F\_d = E$ where $\dim F\_k = k$. When $E$ is equipped with a positive definite symmetric or Hermitian form, this is the same as a splitting; the summands of the splitting are $F\_k \cap F\_{k-1}^{\perp}$. 


The flag formulation works better when working with holomorphic vector bundles, in which case the statement is that the vector bundle has a filtration with one dimensional filtered pieces, not necessarily a splitting.


