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 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[edit]

  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)

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