Optimal approximation constants for varepsilon -isometries ofcontinuous functions spaces

Let X be a Banach space, T be a compact Hausdorff space, C(T) be the real Banach space of all real-valued continuous functions on T endowed with the supremum norm and f:X→ C(T) be an varepsilon-isometry with f(0)=0. Then there exist a nonempty closed subset S⊆ T and a linear isometry g:X→ øverline<roman>span</roman>f(X)S⊆ C(S), g(u)≡ łimn→∞fracf(2^nu)S2^n ∀ u∈ X, such that