Dual norm

The concept of a dual norm arises in functional analysis, a branch of mathematics.

Let $X$ be a normed space (or, in a special case, a Banach space) over a number field $F$ (i.e. $F={\mathbb C}$ or $F={\mathbb R}$) with norm $\left\| \cdot \right\|$. Then the dual (or conjugate) normed space $X'$ (another notation $X^*$) is defined as the set of all continuous linear functionals from $X$ into the base field $F$. If $f:X\to F$ is such a linear functional, then the dual norm^[1] $\|\cdot\|'$ of $f$ is defined by

$\left\| f \right\| ' = \sup\{ \left| f(x) \right| : x \in X, \left\| x \right\| \leq 1\} = \sup \left\{ \frac{ \left| f(x) \right| }{ \left\| x \right\| }: x \in X, x \ne 0 \right\} .$

With this norm, the dual space $X'$ is also a normed space, and moreover a Banach space, since $X'$ is always complete.^[2]

Examples

1. Dual Norm of Vectors

   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.

   For $\sqrt{x^{\mathrm{T}}Qx}$, the dual norm is $\sqrt{y^{\mathrm{T}}Q^{-1}y}$ with $Q$ positive definite.

2. Dual Norm of Matrices

   Frobenius norm

   $\left\| A \right\| _{\text{F}} = \sqrt{\sum_{i=1}^m\sum_{j=1}^n \left| a_{ij} \right| ^2} = \sqrt{\operatorname{trace}(A^{{}^*}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}} \sigma_{i}^2}$

   Its dual norm is $\left\| B \right\| _{\text{F}}$

   Singular value norm

   $\left\| A \right\| _2 = \sigma_{max}(A)$

   Its dual norm is $\sum_i \sigma_i(B)$

Notes

1. A.N.Kolmogorov, S.V.Fomin (1957, III §23)
2. http://www.seas.ucla.edu/~vandenbe/236C/lectures/proxop.pdf

References

• Kolmogorov, A.N.; Fomin, S.V. (1957), Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces, Rochester: Graylock Press
• Rudin, Walter (1991), Functional analysis, McGraw-Hill Science, ISBN 978-0-07-054236-5