How does the distortion of linear embedding of into spaces depend on the height of ?

Let C 0 ( K ) denote the space of all continuous scalar-valued functions defined on the locally compact Hausdorff space K which vanish at infinity, provided with the supremum norm. Let Γ be an infinite set endowed with discrete topology and X a Banach space. We denote by C 0 ( Γ , X ) the Banach space of X -valued functions defined on Γ which vanish at infinity, provided with the supremum norm. In this paper, we prove that, if X has non-trivial cotype and there exists a linear isomorphism T from C 0 ( K ) into C 0 ( Γ , X ) , then K has finite height h t ( K ) , and the distortion ‖ T ‖ ‖ T − 1 ‖ is greater than or equal to 2 h t ( K ) − 1 . The statement of this theorem is optimal and improves a 1970 result of Gordon.