Norm | Sahithyan's Notes

Aka. induced norm.

||x||=\sqrt{\langle x,x\rangle}

A norm is usually associated with an inner product of a vector space. However any arbitrary operation with the below properties can also be defined as a norm:

$||x||: V \rightarrow F$ is a function
$||x|| \ge 0$ and $||x|| = 0 \iff x = \underline{0}$
$||ax|| = |a|\cdot||x||$
$||x+y|| \le ||x|| + ||y||$

Properties

Multiplication

||ax|| = |a| \cdot ||x||

Expansion

|| ax + by || = |a|^2 ||x||^2 + 2\text{Re}(\overline{a}b\langle x,y\rangle) + |b|^2 ||y||^2

Triangle Inequality

|| x + y || \le ||x|| + ||y||

Cauchy-Schwarz Inequality

||\langle x,y\rangle|| \le ||x|| \cdot ||y||

Normed Space

A vector space with a norm.

Distance

For 2 vectors, $x,y$ , distance between the 2 vectors is given by $||x-y||$ .