Dual norm

1 Dual norm

p-norm and q-norm are dual if this holds

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

\Vert y\Vert _\star = \sup\limits_{\Vert x\Vert \leq 1} \langle y,x\rangle

The intuition for the finite-dimensional 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. \Vert y\Vert _* = \sup\limits_{x \neq 0} \dfrac{\langle y,x\rangle}{\Vert x\Vert }.

2 Properties

  • One can easily define the dual norm as:

    \Vert x\Vert _* = \sup\limits_{y \neq 0} \dfrac{\langle y,x\rangle}{\Vert y\Vert }

  • The dual norm is also a norm itself

  • For any x \in E, y \in E^*: x^\top y \leq \Vert x\Vert \cdot \Vert y\Vert _*

  • \left(\Vert x\Vert _p\right)_* = \Vert x\Vert _q if \dfrac{1}{p} + \dfrac{1}{q} = 1, where p, q \geq 1

Example

The Euclidian norm is self dual \left(\Vert x\Vert_2\right)_\star = \Vert x\Vert _2.

3 Examples

Example

Let f(x) = \Vert x\Vert. Prove that f^\star(y) = \mathbb{O}_{\Vert y\Vert _\star \leq 1}