Dual norm

Let be the norm in the primal space , then the following expression defines dual norm:

The intuition for the finite-dimension space is how the linear function (element of the dual space) could stretch the elements of the primal space with respect to their size, i.e.

Properties

  • One can easily define the dual norm as:

  • The dual norm is also a norm itself
  • For any :
  • if , where

Examples

  • Let , then
  • The Euclidian norm is self dual .