An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"
英文摘要：
The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $Δ$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(Δ, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(Δ, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(Δ, X)$ if $K$ is a metric compactum of finite covering dimension.
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