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''linear form, on a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592401.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592402.png" />''
 
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592403.png" /> such that
+
{{MSC|46Exx,|47Axx}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592404.png" /></td> </tr></table>
 
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592406.png" />. The concept of a linear functional, as an important special case of the concept of a [[Linear operator|linear operator]], is one of the main concepts in linear algebra and plays a significant role in analysis.
+
A ''linear functional'', or a
 +
''linear form'', on a [[vector space]] $L$ over a [[field]] $k$ is a
 +
mapping $f:L\to k$ such that
 +
$$\def\l{\lambda} f(x+y) = f(x)+f(y), f(\l x) = \l f(x),$$
 +
for all $x,y\in L$, $\l \in k$. The concept of a
 +
linear functional, as an important special case of the concept of a
 +
[[Linear operator|linear operator]], is one of the main concepts in
 +
linear algebra and plays a significant role in analysis.
  
On the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592407.png" /> of linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592408.png" /> the operations of addition and multiplication by a scalar are defined according to the formulas
+
On the set $L^\#$ of linear functionals on $L$ the operations of addition
 +
and multiplication by a scalar are defined according to the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l0592409.png" /></td> </tr></table>
+
$$(f+g)(x) = f(x) + g(x), (\l f)(x) = \l f(x),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924010.png" /></td> </tr></table>
+
$$f,g\in L^\#,\quad x\in L,\quad \l\in k.$$
 +
They specify in $L^\#$ a vector space structure over $k$, the ''dual space'' or ''linear dual'' of $L$.
  
They specify in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924011.png" /> a vector space structure over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924012.png" />.
+
The kernel of a linear functional is the subspace $\ker f = \{x\in L: f(x)=0\}$. If $f\ne 0 \in L^\#$ (that
 +
is, $f(x) \not\equiv 0\in k$), then $\ker f$ is a hyperplane in $L$. Linear functionals with the
 +
same kernel are proportional.
  
The kernel of a linear functional is the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924014.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924015.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924016.png" /> is a hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924017.png" />. Linear functionals with the same kernel are proportional.
+
If $\{e_\nu : \nu \in \def\L{\Lambda} \L\}$ is a basis of $L$, then for
 +
$$x=\sum_{i=1}^n\l_{\nu_i}e_{\nu_i},\quad \l_{\nu_i}\in k,\quad
 +
f(x)=\sum_{i=1}^n\l_{\nu_i}f(e_{\nu_i}).$$
 +
The correspondence $f\to \{f(x_\nu): \nu\in\L\}$ is an
 +
isomorphism of $L^\#$ onto $k^\L$. Corollary: $L$ is isomorphic to $L^\#$ if
 +
and only if it is finite dimensional. On transition to a new basis in
 +
$L$ the elements $f(e_\nu)\in k$ are transformed by the same formulas as the basis
 +
vectors.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924018.png" /> is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924019.png" />, then for
+
The operator $Q_L:L\to (L^\#)^\#$ defined by $Q_Lx(f) = f(x)$ is injective. It is an isomorphism if
 +
and only if $L$ is finite dimensional. This isomorphism, in contrast
 +
to the isomorphism between $L$ and $L^\#$, is natural, i.e. functorial
 +
(cf.
 +
[[Functorial morphism|Functorial morphism]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924020.png" /></td> </tr></table>
+
A linear functional on a
 +
[[Locally convex space|locally convex space]], in particular on a
 +
normed space, is an important object of study in functional
 +
analysis. Every continuous (as a mapping on topological spaces) linear
 +
functional $f$ on a locally convex space $E$ is bounded (cf.
 +
[[Bounded operator|Bounded operator]]), that is,
 +
$$ \sup_{x\in M} |f(x)| < \infty$$
 +
for all bounded
 +
$M\subset E$. If $E$ is a
 +
[[Normed space|normed space]], the converse is also true; both
 +
properties are then equivalent to the finiteness of the number
 +
$$\|f\| = \sup \{| f(x) | : \|x\|\le 1\}.$$
 +
The continuous linear functionals on a locally convex space $E$ form a
 +
subspace $E^*$ of $E^\#$, which is said to be the dual of $E$. In $E^*$ one
 +
considers different topologies, including the weak and strong
 +
topologies, which correspond, respectively, to pointwise and uniform
 +
convergence on bounded sets. If $E$ is a normed space, then $E^*$ is a
 +
[[Banach space|Banach space]] with respect to the norm $\|f\|$ and the
 +
corresponding topology coincides with the strong topology. The unit
 +
ball $\{f:\|f\|\le 1\}$, considered in the weak topology, is compact.
  
The correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924021.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924022.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924023.png" />. Corollary: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924024.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924025.png" /> if and only if it is finite dimensional. On transition to a new basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924026.png" /> the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924027.png" /> are transformed by the same formulas as the basis vectors.
+
The
 +
[[Hahn–Banach theorem|Hahn–Banach theorem]] has important applications
 +
in analysis; one formulation of it is as follows: If $\|.\|$ is a
 +
[[Pre-norm|pre-norm]] on a vector space $E$ and if $f_0$ is a linear
 +
functional defined on a subspace $E_0$ of $E$ such that $|f_0(x)|\le \|x\|$ for all $x\in E_0$,
 +
then $f_0$ can be extended to the whole of $E$, preserving linearity and
 +
the given bound. Corollary: Any continuous linear functional defined
 +
on a subspace $E_0$ of a locally convex space $E$ can be extended to a
 +
continuous linear functional on $E$, and if $E$ is a normed space,
 +
then the norm is preserved. Hence, for every $x\in E$, $x\ne 0$, there is an $f\in E$
 +
with $f(x)\ne 0$.
  
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924028.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924029.png" /> is injective. It is an isomorphism if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924030.png" /> is finite dimensional. This isomorphism, in contrast to the isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924032.png" />, is natural, i.e. functorial (cf. [[Functorial morphism|Functorial morphism]]).
+
Let $E$ be a normed space and suppose that $E^*$, and then $(E^*)^*$, are
 +
taken with the corresponding norms. Then the operator  
 +
$$R_E:E\to (E^*)^*,\quad R_E x(F) = f(x)$$
 +
is an
 +
isometric imbedding. If under this imbedding $E$ coincides with $(E^*)^*$,
 +
then $E$, which is necessarily complete, is said to be reflexive (cf.
 +
[[Reflexive space|Reflexive space]]). For example, $L_p[a,b]$ and $l_p$, $1\le p<\infty$,
 +
are reflexive if and only if $p>1$. There is a similar concept of
 +
reflexivity for general locally convex spaces.
  
A linear functional on a [[Locally convex space|locally convex space]], in particular on a normed space, is an important object of study in functional analysis. Every continuous (as a mapping on topological spaces) linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924033.png" /> on a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924034.png" /> is bounded (cf. [[Bounded operator|Bounded operator]]), that is,
+
For many locally convex spaces, all linear functionals have been
 
+
described. For example, the adjoint of a Hilbert space $H$ is $\{f:f(x)=(x,x_0) \textrm{ for a fixed } x_0\in H\}$. The
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924035.png" /></td> </tr></table>
+
adjoint of $C[a,b]$ is $\{f:f(x) = \int_a^b x(t)d\mu(t) \textrm{ for a fixed function of bounded variation } \mu(t)\}$.
 
 
for all bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924037.png" /> is a [[Normed space|normed space]], the converse is also true; both properties are then equivalent to the finiteness of the number
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924038.png" /></td> </tr></table>
 
 
 
The continuous linear functionals on a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924039.png" /> form a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924041.png" />, which is said to be the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924042.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924043.png" /> one considers different topologies, including the weak and strong topologies, which correspond, respectively, to pointwise and uniform convergence on bounded sets. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924044.png" /> is a normed space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924045.png" /> is a [[Banach space|Banach space]] with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924046.png" /> and the corresponding topology coincides with the strong topology. The unit ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924047.png" />, considered in the weak topology, is compact.
 
 
 
The [[Hahn–Banach theorem|Hahn–Banach theorem]] has important applications in analysis; one formulation of it is as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924048.png" /> is a [[Pre-norm|pre-norm]] on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924049.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924050.png" /> is a linear functional defined on a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924051.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924052.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924053.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924054.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924055.png" /> can be extended to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924056.png" />, preserving linearity and the given bound. Corollary: Any continuous linear functional defined on a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924057.png" /> of a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924058.png" /> can be extended to a continuous linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924059.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924060.png" /> is a normed space, then the norm is preserved. Hence, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924062.png" />, there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924063.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924064.png" />.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924065.png" /> be a normed space and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924066.png" />, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924067.png" />, are taken with the corresponding norms. Then the operator
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924068.png" /></td> </tr></table>
 
 
 
is an isometric imbedding. If under this imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924069.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924070.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924071.png" />, which is necessarily complete, is said to be reflexive (cf. [[Reflexive space|Reflexive space]]). For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924074.png" />, are reflexive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924075.png" />. There is a similar concept of reflexivity for general locally convex spaces.
 
 
 
For many locally convex spaces, all linear functionals have been described. For example, the adjoint of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924076.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924077.png" />. The adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924078.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059240/l05924079.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N. Bourbaki,   "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian)</TD></TR></table>
+
{|
 
+
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", '''2''', Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)   {{MR|0049861}}
 +
|-
 +
|valign="top"|{{Ref|KoFo}}||valign="top"| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis", '''1–2''', Graylock (1957–1961) (Translated from Russian) {{MR|0085462}} 
 +
|-
 +
|valign="top"|{{Ref|TaLa}}||valign="top"| A.E. Taylor, D.C. Lay, "Introduction to functional analysis", Wiley (1980)  {{MR|0564653}}  {{ZBL|0501.46003}}
 +
|-
 +
|}
  
 
+
[[Category:Linear and multilinear algebra; matrix theory]]
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>
 

Revision as of 22:42, 31 October 2014

2020 Mathematics Subject Classification: Primary: 46Exx, Secondary: 47Axx [MSN][ZBL]


A linear functional, or a linear form, on a vector space $L$ over a field $k$ is a mapping $f:L\to k$ such that $$\def\l{\lambda} f(x+y) = f(x)+f(y), f(\l x) = \l f(x),$$ for all $x,y\in L$, $\l \in k$. The concept of a linear functional, as an important special case of the concept of a linear operator, is one of the main concepts in linear algebra and plays a significant role in analysis.

On the set $L^\#$ of linear functionals on $L$ the operations of addition and multiplication by a scalar are defined according to the formulas

$$(f+g)(x) = f(x) + g(x), (\l f)(x) = \l f(x),$$

$$f,g\in L^\#,\quad x\in L,\quad \l\in k.$$ They specify in $L^\#$ a vector space structure over $k$, the dual space or linear dual of $L$.

The kernel of a linear functional is the subspace $\ker f = \{x\in L: f(x)=0\}$. If $f\ne 0 \in L^\#$ (that is, $f(x) \not\equiv 0\in k$), then $\ker f$ is a hyperplane in $L$. Linear functionals with the same kernel are proportional.

If $\{e_\nu : \nu \in \def\L{\Lambda} \L\}$ is a basis of $L$, then for $$x=\sum_{i=1}^n\l_{\nu_i}e_{\nu_i},\quad \l_{\nu_i}\in k,\quad f(x)=\sum_{i=1}^n\l_{\nu_i}f(e_{\nu_i}).$$ The correspondence $f\to \{f(x_\nu): \nu\in\L\}$ is an isomorphism of $L^\#$ onto $k^\L$. Corollary: $L$ is isomorphic to $L^\#$ if and only if it is finite dimensional. On transition to a new basis in $L$ the elements $f(e_\nu)\in k$ are transformed by the same formulas as the basis vectors.

The operator $Q_L:L\to (L^\#)^\#$ defined by $Q_Lx(f) = f(x)$ is injective. It is an isomorphism if and only if $L$ is finite dimensional. This isomorphism, in contrast to the isomorphism between $L$ and $L^\#$, is natural, i.e. functorial (cf. Functorial morphism).

A linear functional on a locally convex space, in particular on a normed space, is an important object of study in functional analysis. Every continuous (as a mapping on topological spaces) linear functional $f$ on a locally convex space $E$ is bounded (cf. Bounded operator), that is, $$ \sup_{x\in M} |f(x)| < \infty$$ for all bounded $M\subset E$. If $E$ is a normed space, the converse is also true; both properties are then equivalent to the finiteness of the number $$\|f\| = \sup \{| f(x) | : \|x\|\le 1\}.$$ The continuous linear functionals on a locally convex space $E$ form a subspace $E^*$ of $E^\#$, which is said to be the dual of $E$. In $E^*$ one considers different topologies, including the weak and strong topologies, which correspond, respectively, to pointwise and uniform convergence on bounded sets. If $E$ is a normed space, then $E^*$ is a Banach space with respect to the norm $\|f\|$ and the corresponding topology coincides with the strong topology. The unit ball $\{f:\|f\|\le 1\}$, considered in the weak topology, is compact.

The Hahn–Banach theorem has important applications in analysis; one formulation of it is as follows: If $\|.\|$ is a pre-norm on a vector space $E$ and if $f_0$ is a linear functional defined on a subspace $E_0$ of $E$ such that $|f_0(x)|\le \|x\|$ for all $x\in E_0$, then $f_0$ can be extended to the whole of $E$, preserving linearity and the given bound. Corollary: Any continuous linear functional defined on a subspace $E_0$ of a locally convex space $E$ can be extended to a continuous linear functional on $E$, and if $E$ is a normed space, then the norm is preserved. Hence, for every $x\in E$, $x\ne 0$, there is an $f\in E$ with $f(x)\ne 0$.

Let $E$ be a normed space and suppose that $E^*$, and then $(E^*)^*$, are taken with the corresponding norms. Then the operator $$R_E:E\to (E^*)^*,\quad R_E x(F) = f(x)$$ is an isometric imbedding. If under this imbedding $E$ coincides with $(E^*)^*$, then $E$, which is necessarily complete, is said to be reflexive (cf. Reflexive space). For example, $L_p[a,b]$ and $l_p$, $1\le p<\infty$, are reflexive if and only if $p>1$. There is a similar concept of reflexivity for general locally convex spaces.

For many locally convex spaces, all linear functionals have been described. For example, the adjoint of a Hilbert space $H$ is $\{f:f(x)=(x,x_0) \textrm{ for a fixed } x_0\in H\}$. The adjoint of $C[a,b]$ is $\{f:f(x) = \int_a^b x(t)d\mu(t) \textrm{ for a fixed function of bounded variation } \mu(t)\}$.

References

[Bo] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", 2, Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR0049861
[KoFo] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis", 1–2, Graylock (1957–1961) (Translated from Russian) MR0085462
[TaLa] A.E. Taylor, D.C. Lay, "Introduction to functional analysis", Wiley (1980) MR0564653 Zbl 0501.46003
How to Cite This Entry:
Linear functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_functional&oldid=14314
This article was adapted from an original article by A.Ya. Khelemskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article