The concept of a dual norm arises in functional analysis, a branch of mathematics.
Let X be a Banach space with norm \\cdot\. Then the dual space X* is the collection of all continuous linear functionals from X into the base field (which is either R or C). If L is such a linear functional, then the dual norm of L is defined by
 \L\=\sup\{L(x): x\in X, \x\\leq 1\}.
With this norm, the dual space is also a Banach space.
For example, if p, q [1, \infty) satisfy 1/p+1/q=1, then the ^{p} and ^{q} norms are dual to each other. In particular the Euclidean norm is selfdual (p=q=2). Similarly, the Schatten pnorm on matrices is dual to the Schatten qnorm.
