Triangle Inequality

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Theorem

Real Numbers

Let $x, y \in \R$ be real numbers.

Let $\left\vert{x}\right\vert$ be the absolute value of $x$.


Then:

$\left\vert{x + y}\right\vert \le \left\vert{x}\right\vert + \left\vert{y}\right\vert$


Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\left\vert{z}\right\vert$ be the modulus of $z$.


Then:

$\left\vert{z_1 + z_2}\right\vert \le \left\vert{z_1}\right\vert + \left\vert{z_2}\right\vert$


Vectors in $\R^n$

Let $\mathbf{x}$,$\mathbf{y}$ be vectors in $\R^n$.

Let $\left\Vert{\cdot}\right\Vert$ denote vector length.

Then:

$\left \Vert {\mathbf{x} + \mathbf{y} }\right \Vert \le \left \Vert {\mathbf{x}}\right \Vert + \left \Vert { \mathbf{y} }\right \Vert$

If the two vectors are scalar multiples where said scalar is non-negative, an equality holds:

$\exists \lambda \in \R, \lambda \ge 0: \mathbf x = \lambda \mathbf y \iff \left \Vert {\mathbf x + \mathbf y } \right \Vert = \left \Vert { \mathbf x } \right \Vert + \left \Vert { \mathbf y } \right \Vert$


Reverse Triangle Inequality

Let $M = \left({X, d}\right)$ be a metric space.


Then:

$\forall x, y, z \in X: \left|{d \left({x, z}\right) - d \left({y, z}\right)}\right| \le d \left({x, y}\right)$


Normed Vector Spaces

Let $\left({X, \left\lVert{\cdot}\right\rVert}\right)$ be a normed vector space.

Then:

$\forall x, y \in X: \left\lVert{x - y}\right\rVert \ge \big\lvert{\left\lVert{x}\right\rVert - \left\lVert{y}\right\rVert}\big\rvert$


Real and Complex Numbers

Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$.

Then:

$\left\vert{x - y}\right\vert \ge \left\vert{\left\vert{x}\right\vert - \left\vert{y}\right\vert}\right\vert$


Also see

It is in fact a special case of triangle inequality in the Euclidean metric space.


Sources

  • For a video presentation of the contents of this page, visit the Khan Academy.