Dual norm

Let ${\displaystyle X}$ be a normed vector space with norm ${\displaystyle \|\cdot \|}$ and let ${\displaystyle X^{*}}$ be the dual space. The dual norm of a continuous linear functional ${\displaystyle f}$ belonging to ${\displaystyle X^{*}}$ is the non-negative real number defined[1] by any of the following equivalent formulas:

${\displaystyle {\begin{alignedat}{5}\|f\|&=\sup &&\{|f(x)|&&~:~\|x\|\leq 1~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{|f(x)|&&~:~\|x\|<1~&&~{\text{ and }}~&&x\in X\}\\&=\inf &&\{c\in [0,\infty )&&~:~|f(x)|\leq c\|x\|~&&~{\text{ for all }}~&&x\in X\}\\&=\sup &&\{|f(x)|&&~:~\|x\|=1{\text{ or }}0~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{|f(x)|&&~:~\|x\|=1~&&~{\text{ and }}~&&x\in X\}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\&=\sup &&{\bigg \{}{\frac {|f(x)|}{\|x\|}}~&&~:~x\neq 0&&~{\text{ and }}~&&x\in X{\bigg \}}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\\end{alignedat}}}$

where ${\displaystyle \sup }$ and ${\displaystyle \inf }$ denote the supremum and infimum, respectively. The constant 0 map always has norm equal to 0 and it is the origin of the vector space ${\displaystyle X^{*}.}$ If ${\displaystyle X=\{0\}}$ then the only linear functional on ${\displaystyle X}$ is the constant 0 map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal ∞ instead of the correct value of 0.

The map ${\displaystyle f\mapsto \|f\|}$ defines a norm on ${\displaystyle X^{*}.}$ (See Theorems 1 and 2 below.)

The dual norm is a special case of the operator norm defined for each (bounded) linear map between normed vector spaces.

The topology on ${\displaystyle X^{*}}$ induced by ${\displaystyle \|\cdot \|}$ turns out to be as strong as the weak-* topology on ${\displaystyle X^{*}.}$

If the ground field of ${\displaystyle X}$ is complete then ${\displaystyle X^{*}}$ is a Banach space.

The double dual (or second dual) ${\displaystyle X^{**}}$ of ${\displaystyle X}$ is the dual of the normed vector space ${\displaystyle X^{*}}$. There is a natural map ${\displaystyle \varphi :X\to X^{**}}$. Indeed, for each ${\displaystyle w^{*}}$ in ${\displaystyle X^{*}}$ define

${\displaystyle \varphi (v)(w^{*}):=w^{*}(v).}$

The map ${\displaystyle \varphi }$ is linear, injective, and distance preserving.[2] In particular, if ${\displaystyle X}$ is complete (i.e. a Banach space), then ${\displaystyle \varphi }$ is an isometry onto a closed subspace of ${\displaystyle X^{**}}$.[3]

In general, the map ${\displaystyle \varphi }$ is not surjective. For example, if ${\displaystyle X}$ is the Banach space ${\displaystyle L^{\infty }}$ consisting of bounded functions on the real line with the supremum norm, then the map ${\displaystyle \varphi }$ is not surjective. (See ${\displaystyle L^{p}}$ space). If ${\displaystyle \varphi }$ is surjective, then ${\displaystyle X}$ is said to be a reflexive Banach space. If ${\displaystyle 1<p<\infty ,}$ then the space ${\displaystyle L^{p}}$ is a reflexive Banach space.

Let ${\displaystyle \|\cdot \|}$ be a norm on ${\displaystyle \mathbb {R} ^{n}.}$ The associated dual norm, denoted ${\displaystyle \|\cdot \|_{*},}$ is defined as

${\displaystyle \|z\|_{*}=\sup\{z^{\intercal }x\;|\;\|x\|\leq 1\}.}$

(This can be shown to be a norm.) The dual norm can be interpreted as the operator norm of ${\displaystyle z^{\intercal }}$, interpreted as a ${\displaystyle 1\times n}$ matrix, with the norm ${\displaystyle \|\cdot \|}$ on ${\displaystyle \mathbb {R} ^{n}}$, and the absolute value on ${\displaystyle \mathbb {R} }$:

${\displaystyle \|z\|_{*}=\sup\{|z^{\intercal }x|\;|\;\|x\|\leq 1\}.}$

From the definition of dual norm we have the inequality

${\displaystyle z^{\intercal }x=\|x\|\left(z^{\intercal }{\frac {x}{\|x\|}}\right)\leq \|x\|\|z\|_{*}}$

which holds for all x and z.[4] The dual of the dual norm is the original norm: we have ${\displaystyle \|x\|_{**}=\|x\|}$ for all x. (This need not hold in infinite-dimensional vector spaces.)

The dual of the Euclidean norm is the Euclidean norm, since

${\displaystyle \sup\{z^{\intercal }x\;|\;\|x\|_{2}\leq 1\}=\|z\|_{2}.}$

(This follows from the Cauchy–Schwarz inequality; for nonzero z, the value of x that maximises ${\displaystyle z^{\intercal }x}$ over ${\displaystyle \|x\|_{2}\leq 1}$ is ${\displaystyle {\tfrac {z}{\|z\|_{2}}}}$.)

The dual of the ${\displaystyle \ell _{\infty }}$-norm is the ${\displaystyle \ell _{1}}$-norm:

${\displaystyle \sup\{z^{\intercal }x\;|\;\|x\|_{\infty }\leq 1\}=\sum _{i=1}^{n}|z_{i}|=\|z\|_{1},}$

and the dual of the ${\displaystyle \ell _{1}}$-norm is the ${\displaystyle \ell _{\infty }}$-norm.

More generally, Hölder's inequality shows that the dual of the ${\displaystyle \ell _{p}}$-norm is the ${\displaystyle \ell _{q}}$-norm, where, q satisfies ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$, i.e., ${\displaystyle q={\tfrac {p}{p-1}}.}$

As another example, consider the ${\displaystyle \ell _{2}}$- or spectral norm on ${\displaystyle \mathbb {R} ^{m\times n}}$. The associated dual norm is

${\displaystyle \|Z\|_{2*}=\sup\{\mathrm {\bf {tr}} (Z^{\intercal }X)|\|X\|_{2}\leq 1\},}$

which turns out to be the sum of the singular values,

${\displaystyle \|Z\|_{2*}=\sigma _{1}(Z)+\cdots +\sigma _{r}(Z)=\mathrm {\bf {tr}} ({\sqrt {Z^{\intercal }Z}}),}$

where ${\displaystyle r=\mathrm {\bf {rank}} Z.}$ This norm is sometimes called the nuclear norm.[5]

More generally, let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological vector spaces and let ${\displaystyle L(X,Y)}$[6] be the collection of all bounded linear mappings (or operators) of ${\displaystyle X}$ into ${\displaystyle Y}$. In the case where ${\displaystyle X}$ and ${\displaystyle Y}$ are normed vector spaces, ${\displaystyle L(X,Y)}$ can be given a canonical norm.

When ${\displaystyle Y}$ is a scalar field (i.e. ${\displaystyle Y=\mathbb {C} }$ or ${\displaystyle Y=\mathbb {R} }$) so that ${\displaystyle L(X,Y)}$ is the dual space ${\displaystyle X^{*}}$ of ${\displaystyle X}$.

• convex conjugate
• operator norm
• Lp spaces
• Hölder's inequality

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