I should know the answer to this (and I did some time ago, but have forgotten): If the normed linear spaces $X$ and $Y$ are isometric (there is a bijective map from $X$ to $Y$ that preserves distances), are they linearly isomorphic (there is a continuous linear bijection from $X$ to $Y$ with a continuous inverse)?
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This is true for real vector spaces by the Mazur-Ulam theorem which states that a surjective distance-preserving map of one real normed space onto another is an affine map. Indeed, if $f: X \to Y$ is such a map then $g(x) = f(x) - f(0)$ is a linear and onto isometry. The inverse of $g$ is of course linear and isometric, too, so, in partiular, $X$ and $Y$ are linearly isomorphic.
For a proof of this, one can't do better than refer to Väisälä's recent note which appeared in the Monthly, see here for the paywalled published version. References to the original works are given there.