# Dual norm

Let $\|x\|$ be the norm in the primal space $x \in S \subseteq \mathbb{R}^n$, then the following expression defines dual norm:

The intuition for the finite-dimension space is how the linear function (element of the dual space) $f_y(\cdot)$ could stretch the elements of the primal space with respect to their size, i.e. $\|y\|_* = \sup\limits_{x \neq 0} \dfrac{\langle y,x\rangle}{\|x\|}$

# Properties

• One can easily define the dual norm as:

• The dual norm is also a norm itself
• For any $x \in E, y \in E^*$: $x^\top y \leq \|x\| \cdot \|y\|_*$
• $\left(\|x\|_p\right)_* = \|x\|_q$ if $\dfrac{1}{p} + \dfrac{1}{q} = 1$, where $p, q \geq 1$

# Examples

• Let $f(x) = \|x\|$, then $f^*(y) = \mathbb{O}_{\|y\|_* \leq 1}$
• The Euclidian norm is self dual $\left(\|x\|_2\right)_* = \|x\|_2$.