Stability of ε-isometric embeddings into Banach spaces of continuous functions

Let X be a Banach space, T be a compact Hausdorff space and C(T) be the real Banach space of all continuous functions on T endowed with the supremum norm. We show that if there exists a standard ε-isometric embedding f:X→C(T), then there are nonempty closed subset S⊆T and a linearly isometric embedding g:X→span‾(f(X)|S)⊂C(S) defined as g(u)=limn→∞⁡f(2nu)|S2n for each u∈X satisfying that‖f(u)|S−g(u)‖≤4εfor all u∈X. Making use of this result and the well known simultaneous extension operator E:C(S)→C(T), we also prove that the existence of a standard ε-isometric embedding f:X→C(T) implies the existence of a linearly isometric embedding E∘g:X→span‾(E(f(X)|S))⊆C(T) whenever T is metrizable. These conclusions generalize several well-known results. For any compact Hausdorff space (resp. compact metric space) T, we further obtain that if g(X) is complemented in span‾(f(X)|S) (resp. E∘g(X) is complemented in span‾(E(f(X)|S))), then the standard ε-isometric embedding f:X→C(T) is stable.