Given a normed linear space $M$, we can define a metric as $d(x,y) = \left\lVert x-y \right\rVert$, but given a metric $d$, can't we define a norm as: $\left\lVert x \right\rVert = d(0,x)$? Could someone clarify this for me?
2026-04-11 21:38:48.1775943528
Why can't a norm be induced by a metric?
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For a norm, the space $M$ must be a $\Bbb R$-vector space to begin with. Also, we want $\|cx\|=|c|\cdot\|x\|$, but in general $d(0,cx)\ne |c|d(0,x)$.