# 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 ${\mathbb F}$ (i.e. ${\mathbb F}={\mathbb C}$ or ${\mathbb F}={\mathbb R}$) 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 ${\mathbb F}$. If $f:X\to{\mathbb F}$ 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 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.

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

2). Dual Norm of Matrices

Frobenius norm

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

It is dual norm is $\|B\|_F$

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