View source for Dual norm

{{Short description|Measurement on a normed vector space}}
In [[functional analysis]], the '''dual norm''' is a measure of size for a [[continuous function|continuous]] [[linear function]] defined on a [[normed vector space]].

==Definition==

Let <math>X</math> be a [[normed vector space]] with norm <math>\|\cdot\|</math> and let <math>X^*</math> denote its [[continuous dual space]]. The '''dual norm''' of a continuous [[linear functional]] <math>f</math> belonging to <math>X^*</math> is the non-negative real number defined<ref>{{harvnb|Rudin|1991|loc= p. 87}}</ref> by any of the following equivalent formulas:
<math display=block>
\begin{alignat}{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{alignat}
</math>
where <math>\sup</math> and <math>\inf</math> denote the [[supremum and infimum]], respectively. 
The constant <math>0</math> map is the origin of the vector space <math>X^*</math> and it always has norm <math>\|0\| = 0.</math> 
If <math>X = \{0\}</math> then the only linear functional on <math>X</math> is the constant <math>0</math> map and moreover, the sets in the last two rows will both be empty and consequently, their [[supremum]]s will equal <math>\sup \varnothing = - \infty</math> instead of the correct value of <math>0.</math>

Importantly, a linear function <math>f</math> is not, in general, guaranteed to achieve its norm <math>\|f\| = \sup \{|f (x)| : \|x\| \leq 1, x \in X\}</math> on the closed unit ball <math>\{x \in X : \|x\| \leq 1\},</math> meaning that there might not exist any vector <math>u \in X</math> of norm <math>\|u\| \leq 1</math> such that <math>\|f\| = |f u|</math> (if such a vector does exist and if <math>f \neq 0,</math> then <math>u</math> would necessarily have unit norm <math>\|u\| = 1</math>). 
R.C. James proved [[James's theorem]] in 1964, which states that a [[Banach space]] <math>X</math> is [[reflexive space|reflexive]] if and only if every bounded linear function <math>f \in X^*</math> achieves its norm on the closed unit ball.{{sfn|Diestel|1984|p=6}} 
It follows, in particular, that every non-reflexive Banach space has some bounded linear functional that does not achieve its norm on the closed unit ball. 
However, the [[Bishop–Phelps theorem]] guarantees that the set of bounded linear functionals that achieve their norm on the unit sphere of a [[Banach space]] is a norm-[[Dense set|dense subset]] of the [[continuous dual space]].<ref name="BishopPhelps1961">{{cite journal|last1=Bishop|first1=Errett|author-link1=Errett Bishop|last2=Phelps|first2=R. R.|author-link2=Robert R. Phelps|title=A proof that every Banach space is subreflexive|journal=Bulletin of the American Mathematical Society|volume=67|year=1961|pages=97–98|mr=123174|doi=10.1090/s0002-9904-1961-10514-4|doi-access=free}}</ref><ref name="Lomonosov2000">{{cite journal|last1=Lomonosov|first1=Victor|author-link1=Victor Lomonosov|title=A counterexample to the Bishop-Phelps theorem in complex spaces|journal=[[Israel Journal of Mathematics]]|date=2000|volume=115|pages=25–28|doi=10.1007/bf02810578|doi-access=|mr=1749671|s2cid=53646715 }}</ref> 

The map <math>f \mapsto \|f\|</math> defines a [[normed vector space|norm]] on <math>X^*.</math> (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. 
Since the [[ground field]] of <math>X</math> (<math>\Reals</math> or <math>\Complex</math>) is [[Complete metric space|complete]], <math>X^*</math> is a [[Banach space#Linear operators, isomorphisms|Banach space]]. 
The topology on <math>X^*</math> [[normed vector space#Topological structure|induced by]] <math>\|\cdot\|</math> turns out to be stronger than the [[Weak topology#The weak-.2A topology|weak-* topology]] on <math>X^*.</math>

==The double dual of a normed linear space==

The [[double dual]] (or second dual) <math>X^{**}</math> of <math>X</math> is the dual of the normed vector space <math>X^*</math>. There is a natural map <math>\varphi: X \to X^{**}</math>. Indeed, for each <math>w^*</math> in <math>X^*</math> define
<math display=block>\varphi(v)(w^*): = w^*(v).</math>

The map <math>\varphi</math> is [[linear map|linear]], [[injective]], and [[Isometry|distance preserving]].<ref>{{harvnb|Rudin|1991|loc=section 4.5, p. 95}}</ref> In particular, if <math>X</math> is complete (i.e. a Banach space), then <math>\varphi</math> is an isometry onto a closed subspace of <math>X^{**}</math>.<ref>{{harvnb|Rudin|1991|loc=p. 95}}</ref>

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

==Examples==

===Dual norm for matrices===
{{main|Hilbert–Schmidt operator|Matrix norm#Frobenius norm}}

The [[Matrix norm#Frobenius norm|''{{visible anchor|Frobenius norm}}'']] defined by
<math display=block>\| A\|_{\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}</math>
is self-dual, i.e., its dual norm is <math> \| \cdot \|'_{\text{F}} = \| \cdot \|_{\text{F}}.</math>

The ''{{visible anchor|spectral norm}}'', a special case of the [[Matrix norm#induced norm|''induced norm'']] when <math>p=2</math>, is defined by the maximum [[singular value decomposition#Singular values, singular vectors, and their relation to the SVD|singular values]] of a matrix, that is,
<math display=block>\| A \| _2 = \sigma_{\max}(A),</math>
has the nuclear norm as its dual norm, which is defined by 
<math display=block>\|B\|'_2 = \sum_i \sigma_i(B),</math> 
for any matrix <math>B</math> where <math>\sigma_i(B)</math> denote the singular values{{Citation needed|date=March 2018}}.

If <math>p, q \in [1, \infty]</math> the [[Matrix_norm|Schatten <math>\ell^p</math>-norm]] on matrices is dual to the Schatten <math>\ell^q</math>-norm. 

===Finite-dimensional spaces===
Let <math>\|\cdot\|</math> be a norm on <math>\R^n.</math> The associated ''dual norm'', denoted <math>\| \cdot \|_*,</math> is defined as
<math display=block>\|z\|_* = \sup\{z^\intercal x : \|x\| \leq 1 \}.</math>

(This can be shown to be a norm.) The dual norm can be interpreted as the [[operator norm]] of <math>z^\intercal,</math> interpreted as a <math>1 \times n</math> matrix, with the norm <math>\|\cdot\|</math> on <math>\R^n</math>, and the absolute value on <math>\R</math>:
<math display=block>\|z\|_* = \sup\{|z^\intercal x| : \|x\| \leq 1 \}.</math>

From the definition of dual norm we have the inequality
<math display=block>z^\intercal x = \|x\| \left(z^\intercal \frac{x}{\|x\|} \right) \leq \|x\| \|z\|_*</math>
which holds for all <math>x</math> and <math>z.</math><ref>{{harvnb|Boyd|Vandenberghe|2004|loc=[https://web.stanford.edu/~boyd/cvxbook/ p. 637]}}</ref><ref>This inequality is tight, in the following sense: for any <math>x</math> there is a <math>z</math> for which the inequality holds with equality. (Similarly, for any <math>z</math> there is an <math>x</math> that gives equality.)</ref> The dual of the dual norm is the original norm: we have <math>\|x\|_{**} = \|x\| </math> for all <math>x.</math> (This need not hold in infinite-dimensional vector spaces.)

The dual of the [[Norm (mathematics)#Euclidean norm|Euclidean norm]] is the Euclidean norm, since
<math display=block>\sup\{z^\intercal x : \|x\|_2 \leq 1 \} = \|z\|_2.</math>

(This follows from the [[Cauchy–Schwarz inequality]]; for nonzero <math>z,</math> the value of <math>x</math> that maximises <math>z^\intercal x</math> over <math>\|x\|_2 \leq 1</math> is <math>\tfrac{z}{\|z\|_2}.</math>)

The dual of the <math>\ell^\infty </math>-norm is the <math>\ell^1</math>-norm:
<math display=block>\sup\{z^\intercal x : \|x\| _\infty \leq 1\} = \sum_{i=1}^n |z_i| = \|z\| _1,</math>
and the dual of the <math>\ell^1</math>-norm is the <math>\ell^\infty</math>-norm.

More generally, [[Hölder's inequality]] shows that the dual of the [[Lp spaces|<math>\ell^p</math>-norm]] is the <math>\ell^q</math>-norm, where <math>q</math> satisfies <math>\tfrac{1}{p} + \tfrac{1}{q} = 1,</math> that is, <math>q = \tfrac{p}{p-1}.</math>

As another example, consider the <math>\ell^2</math>- or spectral norm on <math>\R^{m\times n}</math>. The associated dual norm is
<math display=block>\|Z\| _{2*} = \sup\{\mathbf{tr}(Z^\intercal X) : \|X\|_2 \leq 1\},</math>
which turns out to be the sum of the singular values,
<math display=block>\|Z\| _{2*} = \sigma_1(Z) + \cdots + \sigma_r(Z) = \mathbf{tr} (\sqrt{Z^\intercal Z}),</math>
where <math>r = \mathbf{rank} Z.</math> This norm is sometimes called the [[nuclear operator|''{{visible anchor|nuclear norm}}'']].<ref>{{harvnb|Boyd|Vandenberghe|2004|loc=[https://web.stanford.edu/~boyd/cvxbook/ p. 637]}}</ref>

===''L<sup>p</sup>'' and ℓ<sup>''p''</sup> spaces===
{{See also|Lp space|Riesz representation theorem}}

For <math>p \in [1, \infty],</math> {{mvar|p}}-norm (also called <math>\ell_p</math>-norm) of vector <math>\mathbf{x} = (x_n)_n</math> is
<math display=block>\|\mathbf{x}\|_p ~:=~ \left(\sum_{i=1}^n \left|x_i\right|^p\right)^{1/p}.</math>

If <math>p, q \in [1, \infty]</math> satisfy <math>1/p+1/q=1</math> then the <math>\ell^p</math> and <math>\ell^q</math> norms are dual to each other and the same is true of the <math>L^p</math> and <math>L^q</math> norms, where <math>(X, \Sigma, \mu),</math> is some [[measure (mathematics)|measure space]]. 
In particular the [[L2 norm|Euclidean norm]] is self-dual since <math>p = q = 2.</math> 
For <math>\sqrt{x^{\mathrm{T}}Qx}</math>, the dual norm is <math>\sqrt{y^{\mathrm{T}}Q^{-1}y}</math> with <math>Q</math> positive definite.

For <math>p = 2,</math> the <math>\|\,\cdot\,\|_2</math>-norm is even induced by a canonical [[inner product]] <math>\langle \,\cdot,\,\cdot\rangle,</math> meaning that <math>\|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}</math> for all vectors <math>\mathbf{x}.</math> This inner product can expressed in terms of the norm by using the [[polarization identity]]. 
On <math>\ell^2,</math> this is the ''{{visible anchor|Euclidean inner product}}'' defined by
<math display=block>\langle \left(x_n\right)_{n}, \left(y_n\right)_{n} \rangle_{\ell^2} ~=~ \sum_n x_n \overline{y_n}</math>
while for the space <math>L^2(X, \mu)</math> associated with a [[measure (mathematics)|measure space]] <math>(X, \Sigma, \mu),</math> which consists of all [[square-integrable function]]s, this inner product is 
<math display=block>\langle f, g \rangle_{L^2} = \int_X f(x) \overline{g(x)} \, \mathrm dx.</math>
The norms of the continuous dual spaces of <math>\ell^2</math> and <math>\ell^2</math> satisfy the [[polarization identity]], and so these dual norms can be used to define inner products. With this inner product, this dual space is also a [[Hilbert space]].

==Properties==

Given normed vector spaces  <math>X</math> and <math>Y,</math> let <math>L(X,Y)</math><ref>Each <math>L(X,Y)</math> is a [[vector space]], with the usual definitions of addition and scalar multiplication of functions; this only depends on the vector space structure of <math>Y</math>, not <math>X</math>.</ref> be the collection of all [[Bounded operator|bounded linear mappings]] (or {{em|operators}}) of <math>X</math> into <math>Y.</math> Then <math>L(X,Y)</math> can be given a canonical norm.

{{Math theorem|name=Theorem 1|math_statement=
Let <math>X</math> and <math>Y</math> be normed spaces. Assigning to each continuous linear operator <math>f \in L(X, Y)</math> the scalar <math display=block>\|f\| = \sup \{\|f(x)\| : x \in X, \|x\| \leq 1\}</math>
defines a norm <math>\|\cdot\| ~:~ L(X, Y) \to \Reals</math> on <math>L(X, Y)</math> that makes <math>L(X, Y)</math> into a normed space. Moreover, if <math>Y</math> is a Banach space then so is <math>L(X, Y).</math><ref>{{harvnb|Rudin|1991|page=92}}</ref>
}}

{{collapse top|title=Proof|left=true}}
A subset of a normed space is bounded [[if and only if]] it lies in some multiple of the [[unit sphere]]; thus <math>\|f\| < \infty</math> for every <math>f \in L(X,Y)</math> if <math>\alpha</math> is a scalar, then <math>(\alpha f)(x) = \alpha \cdot f x</math> so that 
<math display=block>\|\alpha f\| = |\alpha| \|f\|.</math>

The [[triangle inequality]] in <math>Y</math> shows that
<math display=block>\begin{align}
\| \left(f_1 + f_2\right) x \| ~&=~ \|f_1 x + f_2 x\| \\
&\leq~ \|f_1 x\| + \|f_2 x\| \\
&\leq~ \left(\|f_1\| + \|f_2\|\right) \|x\| \\
&\leq~ \|f_1\| + \|f_2\|
\end{align}</math>

for every <math>x \in X</math> satisfying <math>\|x\| \leq 1.</math> This fact together with the definition of <math>\| \cdot \| ~:~ L(X, Y) \to \mathbb{R}</math> implies the triangle inequality:
<math display=block>\|f + g\| \leq \|f\| + \|g\|.</math>

Since <math>\{ |f(x)| : x \in X, \|x\| \leq 1 \}</math> is a non-empty set of non-negative real numbers, <math>\|f\| = \sup \left\{ |f(x)| : x \in X, \| x \| \leq 1 \right\}</math> is a non-negative real number. 
If <math>f \neq 0</math> then <math>f x_0 \neq 0</math> for some <math>x_0 \in X,</math> which implies that <math>\left\|f x_0\right\| > 0</math> and consequently <math>\|f\| > 0.</math> This shows that <math>\left( L(X, Y), \| \cdot \|\right)</math> is a normed space.<ref>{{harvnb|Rudin|1991|page=93}}</ref>

Assume now that <math>Y</math> is complete and we will show that <math>( L(X, Y), \| \cdot \|)</math> is complete. Let <math>f_{\bull} = \left(f_n\right)_{n=1}^{\infty}</math> be a [[Cauchy sequence]] in <math>L(X, Y),</math> so by definition <math>\left\|f_n - f_m\right\| \to 0</math> as <math>n, m \to \infty.</math> This fact together with the relation
<math display=block>\left\|f_n x - f_m x\right\| = \left\| \left( f_n - f_m \right) x \right\| \leq \left\|f_n - f_m\right\| \|x\|</math>

implies that <math>\left(f_nx \right)_{n=1}^{\infty}</math> is a Cauchy sequence in <math>Y</math> for every <math>x \in X.</math> It follows that for every <math>x \in X,</math> the limit <math>\lim_{n \to \infty} f_n x</math> exists in <math>Y</math> and so we will denote this (necessarily unique) limit by <math>f x,</math> that is:
<math display=block>f x ~=~ \lim_{n \to \infty} f_n x.</math>

It can be shown that <math>f: X \to Y</math> is linear. If <math>\varepsilon > 0</math>, then <math>\left\|f_n - f_m\right\|  \| x \| ~\leq~ \varepsilon \|x\|</math> for all sufficiently large integers {{mvar|n}} and {{mvar|m}}. It follows that
<math display=block>\left\|fx - f_m x\right\| ~\leq~ \varepsilon \|x\|</math>
for sufficiently all large <math>m.</math> Hence <math>\|fx\| \leq \left( \left\|f_m\right\| + \varepsilon \right) \|x\|,</math> so that <math>f \in L(X, Y)</math> and <math>\left\|f - f_m\right\| \leq \varepsilon.</math> This shows that <math>f_m \to f</math> in the norm topology of <math>L(X, Y).</math> This establishes the completeness of <math>L(X, Y).</math><ref>{{harvnb|Rudin|1991|page=93}}</ref>
{{collapse bottom}} 

When <math>Y</math> is a [[scalar field]] (i.e. <math>Y = \Complex</math> or <math>Y = \R</math>) so that <math>L(X,Y)</math> is the [[dual space]] <math>X^*</math> of <math>X.</math>

{{Math theorem|name=Theorem 2|math_statement=
Let <math>X</math> be a normed space and for every <math>x^* \in X^*</math> let
<math display=block>\left\|x^*\right\| ~:=~ \sup \left\{| \langle x, x^* \rangle | ~:~ x \in X \text{ with } \| x \| \leq 1 \right\}</math>
where by definition <math>\langle x, x^* \rangle ~:=~ x^{*}(x)</math> is a scalar. 
Then
<ol>
<li><math>\| \, \cdot \, \| : X^* \to \R</math> is a [[Norm (mathematics)|norm]] that makes <math>X^*</math> a Banach space.<ref>{{harvnb|Aliprantis|Border|2006|page=230}}</ref></li>
<li>If <math>B^*</math> is the closed unit ball of <math>X^*</math> then for every <math>x \in X,</math>
<math display=block>\begin{alignat}{4}
\| x \| ~&=~ \sup \left\{ | \langle x, x^* \rangle | ~:~ x^* \in B^* \right\} \\
&=~ \sup \left\{ \left|x^*(x)\right| ~:~ \left\|x^*\right\| \leq 1 \text{ with } x^* \in X^* \right\}. \\
\end{alignat}</math>
Consequently, <math>x^* \mapsto \langle x, x^* \rangle</math> is a bounded [[linear functional]] on <math>X^*</math> with norm <math>\| x^* \| ~=~ \| x \|.</math></li>
<li><math>B^*</math> is weak*-compact.</li>
</ol>
}}

{{collapse top|title=Proof|left=true}}
Let <math>B ~=~ \sup\{ x \in X ~:~ \| x \| \le 1 \}</math>denote the closed unit ball of a normed space <math>X.</math> 
When <math>Y</math> is the [[scalar field]] then <math>L(X,Y) = X^*</math> so part (a) is a corollary of Theorem 1. Fix <math>x \in X.</math> There exists<ref>{{harvnb|Rudin|1991|loc=Theorem 3.3 Corollary, p. 59}}</ref> <math>y^* \in B^*</math> such that
<math display=block>\langle{x,y^*}\rangle = \|x\|.</math>
but,
<math display=block>|\langle{x,x^*}\rangle| \leq \|x\|\|x^*\| \leq \|x\|</math>
for every <math>x^* \in B^*</math>. (b) follows from the above. Since the open unit ball <math>U</math> of <math>X</math> is dense in <math>B</math>, the definition of <math>\|x^*\|</math> shows that <math>x^* \in B^*</math> [[if and only if]] <math>|\langle{x,x^*}\rangle| \leq 1</math> for every <math>x \in U</math>. The proof for (c)<ref>{{harvnb|Rudin|1991|loc=Theorem 3.15 The [[Banach–Alaoglu theorem]] algorithm, p. 68}}</ref> now follows directly.<ref>{{harvnb|Rudin|1991|page=94}}</ref>
{{collapse bottom}} 
<!--The {{em|[[dual space|topological dual]]}} (or {{em|conjugate}}) {{em|normed}} space <math>X^{\prime}</math> is defined as the set of all continuous linear functionals from <math>X</math> into the base field <math>F</math>. Let <math>X</math> and <math>X^{\prime}</math> be normed -->
<!--If <math>f : X \to F</math> is such a linear functional, then the {{em|dual norm}}wrong ref <ref>{{harvtxt|A.N.Kolmogorov, S.V.Fomin|1957|loc=III §23}}</ref> <math>\|\cdot\|'</math> of <math>f</math> is defined by-->
<!--: <math>\|f\| ' = \sup\{ |f(x)| : x \in X, \left\| x \right\| \leq 1\} = \sup \left\{ \frac{ |f(x)| }{ \|x\| }: x \in X, x \neq 0 \right\} .</math>-->

As usual, let <math>d(x, y) := \|x - y\|</math> denote the canonical [[Metric (mathematics)|metric]] induced by the norm on <math>X,</math> and denote the distance from a point <math>x</math> to the subset <math>S \subseteq X</math> by 
<math display=block>d(x, S) ~:=~ \inf_{s \in S} d(x, s) ~=~ \inf_{s \in S} \|x - s\|.</math>
If <math>f</math> is a bounded linear functional on a normed space <math>X,</math> then for every vector <math>x \in X,</math>{{sfn|Hashimoto|Nakamura|Oharu|1986|p=281}}
<math display=block>|f(x)| = \|f\| \, d(x, \ker f),</math>
where <math>\ker f = \{k \in X : f(k) = 0\}</math> denotes the [[Kernel (linear algebra)|kernel]] of <math>f.</math>

==See also==

* {{annotated link|Convex conjugate}}
* {{annotated link|Hölder's inequality}}
* {{annotated link|Lp space}}
* {{annotated link|Operator norm}}
* {{annotated link|Polarization identity}}

==Notes==
{{reflist}}

==References==

* {{cite book | last1=Aliprantis | first1=Charalambos D. | last2= Border | first2=Kim C. | title=Infinite Dimensional Analysis: A Hitchhiker's Guide | publisher=Springer | year=2006 | edition=3rd | isbn=9783540326960}}
* {{cite book | last1=Boyd | first1=Stephen | author-link1=Stephen P. Boyd | last2=Vandenberghe | first2=Lieven | title=Convex Optimization | year= 2004 | publisher=[[Cambridge University Press]] | isbn=9780521833783}}
* {{cite book|last=Diestel|first=Joe|title=Sequences and series in Banach spaces|publisher=Springer-Verlag|publication-place=New York|date=1984|isbn=0-387-90859-5|oclc=9556781}} <!--{{sfn|Diestel|1984|p=}}-->
* {{cite journal|last1=Hashimoto|first1=Kazuo|last2=Nakamura|first2=Gen|last3=Oharu|first3=Shinnosuke|title=Riesz's lemma and orthogonality in normed spaces|journal=Hiroshima Mathematical Journal|publisher=Hiroshima University - Department of Mathematics|volume=16|issue=2|date=1986-01-01|issn=0018-2079|doi=10.32917/hmj/1206130429|url=https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-16/issue-2/Rieszs-lemma-and-orthogonality-in-normed-spaces/10.32917/hmj/1206130429.pdf}} <!--{{sfn|Hashimoto|Nakamura|Oharu|1986|p=}}-->
* {{cite book | last1 = Kolmogorov| first1 = A.N.| author-link1=Andrei Kolmogorov | last2 = Fomin | first2 = S.V. | author-link2=Sergei Fomin | year = 1957 | title = Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces | publisher = Graylock Press | location = Rochester }}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->

==External links==

* [http://www.seas.ucla.edu/~vandenbe/236C/lectures/proxop.pdf Notes on the proximal mapping by Lieven Vandenberge]

{{Banach spaces}}
{{Duality and spaces of linear maps}}
{{Functional analysis}}

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[[Category:Linear algebra]]
[[Category:Mathematical optimization]]
[[Category:Linear functionals]]