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How can one show that a norm-preserving map $T: X \rightarrow X'$ where $X,X'$ are vector spaces and $T(0) = 0$ is linear? Thanks in advance.

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    Could you clarify what you mean by norm-preserving? Note that if $T$ is not linear, then $|Tx| = |x|$ does *not* mean that $|Tx - Ty| = |x - y|$, which is what is usually meant (and as noted below, your claim is false if you only require $|Tx| = |x|$ and true in many cases if you actually meant distance-preserving).2012-05-17
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    Related threads: http://math.stackexchange.com/q/121046, http://math.stackexchange.com/q/81086, http://math.stackexchange.com/q/169652012-05-17

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