Dual norm

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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[edit]

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)

Notes[edit]

References[edit]

  • 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