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A norm $\Vert\cdot\Vert$ on a vector space $V$ is required to satisfy the triangle inequality: $\Vert x+y\Vert\leqslant\Vert x\Vert+\Vert y\Vert.$ The function of this inequality is to ensure that mapping $f:V^2\to V,\ (x,y)\mapsto x+y$ is continuous. But why don't we simply require that $\displaystyle\lim_{(x,y)\to (0,0)} \Vert x+y\Vert=0,$ instead of the subadditivity? Or, can we repalce the triangle inequality with $\Vert x+y\Vert\leqslant 2\Vert x\Vert +3\Vert y\Vert,$ or something like that?

Question: What's the irreplaceability of triangle inequality in the normed vector space theory?

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The function of this inequality is not to have "addition" continuous. The triangular inequality is also required for metric spaces, such as $\mathbb{Q}$ with the discrete metric. I do not think that people are actually interested in discussing continuity of the "+" function.

The triangular inequality is based on a much more basic idea: The norm should represent the length of a vector $x+y$. When you are not measuring "straight" from the beginning $0$ to the end $x+y$, but taking some detour via $x$, you should at least not get a shorter measurement.

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  • $\begingroup$ So the triangle inequality comes from geometric intuition, right? $\endgroup$
    – painday
    May 5, 2018 at 5:19
  • $\begingroup$ In a sense, yes. The three properties for a norm (positive definiteness, homogeneous, triangular inequality) are axioms. These axioms are what people consider to be reasonable as properties for a length measure. $\endgroup$
    – Jonas
    May 5, 2018 at 5:29
  • $\begingroup$ OK, I see. So triangle inequality is responsible for the geometry of the vector space. If we replace this with something else, the measure theory of graphs in the vector space would be rather strange. $\endgroup$
    – painday
    May 5, 2018 at 5:36
  • $\begingroup$ This situation can occur, for instance in an $L^p$ space, with 0 < p < 1. There is also the concept of Quasinorm which is a generalisation of a norm that does not fulfil subadditivity - but a similar property. $\endgroup$
    – Jonas
    May 5, 2018 at 5:43
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As I understand it the triangle inequality derives from metric spaces: "Visually" spoken, the direct connection between two points $A$ and $B$ should never be longer than the sum of the two distances $\overline{AC}$ and $\overline{CB}$ for any other point $C$.

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