When do the C0(1)(K,X) spaces determine the locally compact subspaces K of the real line RR?

Let X be a finite-dimensional space and K   a locally compact subspace of the real line RR without isolated points. The space C0(1)(K,X) will stand for the Banach space of all X-valued continuously differentiable functions f such that f   and f′f′ vanish at infinity, endowed with the norm ‖f‖=max⁡{‖f‖∞,‖f′‖∞}‖f‖=max⁡{‖f‖∞,‖f′‖∞}. It is shown that if there is an isomorphism T:C0(1)(K,X)→C0(1)(S,X) satisfying‖T‖‖T−1‖<λ(X)and‖T‖∞‖T−1‖∞<∞ then K and S   are homeomorphic, where λ(X)λ(X) is a parameter introduced by Jarosz in 1989 and ‖T‖∞‖T‖∞ is the norm of T   when C0(1)(K,X) and C0(1)(S,X) are considered as subspaces of C0(K,X)C0(K,X) and C0(S,X)C0(S,X), respectively.