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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
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.
Dual Norm of Matrices
Frobenius norm
\( \|A\|_{\text{F}}=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2}=\sqrt{\operatorname{trace}(A^{{}^*}A)}=\sqrt{\sum_{i=1}^{\min\{m,\,n\}} \sigma_{i}^2}\)
Its dual norm is\( \|B\|_{\text{F}}\)
Singular value norm
\( \|A\|_2=\sigma_{max}(A)\)
Dual norm \( \sum_i \sigma_i(B)\)
Notes
A.N.Kolmogorov, S.V.Fomin (1957, III §23)
http://www.seas.ucla.edu/~vandenbe/236C/lectures/proxop.pdf
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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