Embedding subspaces of 𝑙ⁿ_{∞} into spaces with Schauder basis

Abstract

It is proved that for sufficiently small ε > 0 \varepsilon > 0 and any 0 > δ > 1 / 2 0 > \delta > 1/2 , a random n n -dimensional subspace E E of l ∞ N l_\infty ^N , where N = ( 1 + ε ) n N = (1 + \varepsilon )n , has the property: whenever E E is embedded into any ( 1 + γ ) n (1 + \gamma )n -dimensional space with a basis, where γ = c δ ε \gamma = c\delta \varepsilon , then the embedding constant exceeds c ′ n 1 / 2 − δ {c’}{n^{1/2 - \delta }} .