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I'm trying to understand what is norm in one dimension. A norm in two dimensions I do understand - it's a length of a vector, which makes sense. But, according to Wikipedia a norm is positive length and $1$-norm is just the absolute value. I don't get the intuition of what the length in one dimension means.

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  • $\begingroup$ Distance between points in the real line $\endgroup$ – janmarqz Jun 8 '18 at 21:03
  • $\begingroup$ In 1 dimension it's the length of vectors just like in 2 dimensions $\endgroup$ – Jakobian Jun 8 '18 at 21:06
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If you understand what a norm is in dimension $2$, then you should know that, if $V$ is a vector space of finite dimension, a norm on $V$ is a function $\lVert {}\cdot{} \rVert \colon V \to \mathbb{R}_{\geq 0}$ which essentially associates to each vector $v \in V$ its length. When $V \cong \mathbb{R}$ you just denote $\lVert {}\cdot{}\rVert$ by $\lvert {}\cdot{}\rvert$. Geometrically, $\lvert x-y \rvert $ is the distance between $x$ and $y$ in $\mathbb{R}$.

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  • $\begingroup$ The vector space doesn't have to have a finite dimension to consider a norm $\endgroup$ – Jakobian Jun 8 '18 at 21:08
  • $\begingroup$ I added that information just because it seems the OP is working in finite dimension. $\endgroup$ – Gibbs Jun 8 '18 at 21:08
  • $\begingroup$ This answer is good, but i think it's too abstract for the OP $\endgroup$ – Jakobian Jun 8 '18 at 21:12
  • $\begingroup$ In case she/he needs some more explanations, she/he can just ask, and I will try to answer. $\endgroup$ – Gibbs Jun 8 '18 at 21:15
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You may think of length of a real number as the distance from the real number zero. For example length of $x=-5$ is the distance from $-5$ to $0$ which is $5$

Note that this is just the absolute value of the point.

Thus you may say norm in one dimension is the same as absolute value.

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