View source for Linear functional


{{MSC|46Exx,|47Axx}}
{{TEX|done}}


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 $L^\#$ 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|Functorial morphism]]).

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
[[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 $f(x)\ne 0$.

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.

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====
{|
|-
|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]]