# Triangle Inequality

## Theorem

### Geometry

Given a triangle $ABC$, the sum of the lengths of any two sides of the triangle is greater than the length of the third side.

In the words of Euclid:

In any triangle two sides taken together in any manner are greater than the remaining one.

### 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 $\cmod z$ denote the modulus of $z$.

Then:

$\cmod {z_1 + z_2} \le \cmod {z_1} + \cmod {z_2}$

### Vectors in $\R^n$

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

Let $\norm {\, \cdot \,}$ denote vector length.

Then:

$\norm {\mathbf x + \mathbf y} \le \norm {\mathbf x} + \norm {\mathbf y}$

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

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

## Triangle Inequality for Integrals

Let $\struct {X, \Sigma, \mu}$ be a measure space.

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

Then:

$\ds \size {\int_X f \rd \mu} \le \int_X \size f \rd \mu$