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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 p and 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.






Source: Wikipedia | The above article is available under the GNU FDL. | Edit this article



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