Non-Banach, completely metrizable normed vector space Does there exist a normed vector space $(X,\|\cdot\|)$ over $\mathbb R$ or $\mathbb C$ such that


*

*the metric induced by the norm $(x,y)\mapsto\|x-y\|$ is not complete; but

*there exists some other metric $d$ on $X$ that is complete and defines the same topology on $X$ as the norm?


It may be possible, since completeness of a metric is not a topological property, but I can't think of an obvious counterexample.
In other words: is every completely metrizable (by some metric) normed vector space necessarily a Banach space (i.e., complete also in the norm metric)?

The answer is yes, any such space must be a Banach space. This result was proved by Victor Klee in 1952 and answered a question first asked by Banach in 1932.

V. L. Klee, Invariant metrics in groups (Solution of a problem of Banach), Proc.
  Amer. Math. Soc., 3 (1952), 484–487.

