Dual norm

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 ℓ^p and ℓ^q norms are dual to each other. In particular the Euclidean norm is self-dual (p=q=2). Similarly, the Schatten p-norm on matrices is dual to the Schatten q-norm.

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