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| The concept of a '''dual norm''' arises in [[functional analysis]], a branch of mathematics.
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| Let <math>X</math> be a [[normed space]] (or, in a special case, a [[Banach space]]) over a number field <math>{\mathbb F}</math> (i.e. <math>{\mathbb F}={\mathbb C}</math> or <math>{\mathbb F}={\mathbb R}</math>) with [[norm (mathematics)|norm]] <math>\|\cdot\|</math>. Then the [[dual space|''dual'']] (or ''conjugate'') ''normed'' space <math>X'</math> (another notation <math>X^*</math>) is defined as the set of all [[continuous function|continuous]] [[linear functional]]s from <math>X</math> into the base field <math>{\mathbb F}</math>. If <math>f:X\to{\mathbb F}</math> is such a linear functional, then the
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| ''dual norm''<ref>{{harvtxt|A.N.Kolmogorov, S.V.Fomin|1957|loc=III §23}}</ref> <math>\|\cdot\|'</math> of <math>f</math> is defined by
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| : <math> \|f\|'=\sup\{|f(x)|: x\in X, \|x\|\leq 1\}=\sup\left\{\frac{|f(x)|}{\|x\|}: x\in X, x\ne 0\right\}. </math>
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| With this norm, the [[dual space]] <math>X'</math> is also a [[normed space]], and moreover a [[Banach space]], since <math>X'</math> is always complete.
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| <ref>http://www.seas.ucla.edu/~vandenbe/236C/lectures/proxop.pdf</ref>
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| == Examples ==
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| 1). '''Dual Norm of Vectors'''
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| If ''p'', ''q'' ∈ <math>[1, \infty]</math> satisfy <math>1/p+1/q=1</math>, then the '''ℓ<sup>p</sup>''' and '''ℓ<sup>q</sup>''' norms are dual to each other.
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| In particular the [[L2 norm|Euclidean norm]] is self-dual (''p''=''q''=2). Similarly, the [[Matrix_norm|Schatten ''p''-norm]] on matrices is dual to the Schatten ''q''-norm.
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| For <math>\sqrt{x^TQx}</math>, the dual norm is <math>\sqrt{y^TQ^{-1}y}</math> with <math>Q</math> positive definite.
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| 2). '''Dual Norm of Matrices'''
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| ''Frobenius norm''
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| :<math>\|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}</math>
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| It is dual norm is <math>\|B\|_F</math>
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| ''Singular value norm''
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| :<math>\|A\|_2=\sigma_{max}(A)</math>
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| Dual norm <math>\sum_i \sigma_i(B)</math>
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{citation
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| | authorlink = A.N.Kolmogorov; S.V.Fomin
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| | last1 = Kolmogorov
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| | first1 = A.N.
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| | last2 = Fomin
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| | first2 = S.V.
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| | year = 1957
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| | title = Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces
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| | publisher = Graylock Press
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| | location = Rochester
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| | isbn =
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| }}
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| * {{citation|authorlink=Walter Rudin|first=Walter|last=Rudin|title=Functional analysis|publisher=McGraw-Hill Science|year=1991|isbn=978-0-07-054236-5}}
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| [[Category:Linear algebra]]
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| {{mathanalysis-stub}}
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