Let $\mathbb F \in \{\mathbb R,\mathbb C\}$. Let $X$ and $Y$ be normed vector spaces over $\mathbb F$, with norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Define metrics $d_X:X\times X\to [0,\infty)$ and $d_Y:Y\times Y\to [0,\infty)$ by
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\begin{align*}
d_X(x,y)=\|x-y\|_X
\end{align*}
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and
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\begin{align*}
d_Y(u,v)=\|u-v\|_Y.
\end{align*}
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Let $T:X\to Y$ be an $\mathbb F$-[linear map](/page/Linear%20Map). Then $T$ is an isometry from $(X,d_X)$ to $(Y,d_Y)$ if and only if