Separable elastic Banach spaces are universal

A Banach space X is elastic if there is a constant K so that whenever a Banach space Y embeds into X, then there is an embedding of Y into X with constant K  . We prove that C[0,1]C[0,1] embeds into separable infinite dimensional elastic Banach spaces, and therefore they are universal for all separable Banach spaces. This confirms a conjecture of Johnson and Odell. The proof uses incremental embeddings into X   of C(K)C(K) spaces for countable compact K   of increasing complexity. To achieve this we develop a generalization of Bourgain's basis index that applies to unconditional sums of Banach spaces and prove a strengthening of the weak injectivity property of these C(K)C(K) that is realized on special reproducible bases.