# 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 $\|\cdot\|$. 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

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

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 ℓp and ℓq 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
$\|A\|_{\text{F}}=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2}=\sqrt{\operatorname{trace}(A^{{}^*}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}} \sigma_{i}^2}$
It is dual norm is $\|B\|_{\text{F}}$
Singular value norm
$\|A\|_2=\sigma_{max}(A)$
Dual norm $\sum_i \sigma_i(B)$

## 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