Spreading model ($\Vert \sum a_i{e}_i\Vert ={\lim }_{n_1\to \infty }\Vert \sum a_i{x}_{n_i}\Vert$)

A normalized basic sequence generates a spreading model when the norms of all finite scalar combinations converge as the first index tends to infinity, uniformly over increasing choices of the indices. The limiting norm on finitely supported scalar sequences is unchanged by increasing relabellings of coordinates. Completing this norm gives a Banach space with its canonical basic sequence. A diagonal application of the finite-colour infinite Ramsey theorem produces such a model from a subsequence of every normalized basic sequence.