Triangle Inequality

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Theorem

Real Numbers

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

Let $\size x$ denote the absolute value of $x$.


Then:

$\size {x + y} \le \size x + \size y$


Complex Numbers

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

Let $\left\vert{z}\right\vert$ denote 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 Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.

Then:

$\forall x, y \in R: \norm {x - y} \ge \big\lvert {\norm x - \norm y} \big\rvert$


Normed Vector Space

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:

$\cmod {x - y} \ge \size {\cmod x - \cmod y}$


Triangle Inequality for Integrals

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \overline{\R}$ be a $\mu$-integrable function.


Then:

$\displaystyle \left\vert{\int_X f \rd \mu}\right\vert \le \int_X \left\vert{f}\right\vert \rd \mu$


Also see

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