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 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
     \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

References

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