Difference between revisions of "Linear functional"
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− | ''linear form, on a [[Vector space|vector space]] | + | ''linear form, on a |
+ | [[Vector space|vector space]] over a field k'' | ||
− | A mapping | + | 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 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. | ||
− | If | + | 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 $\{f:f(x) = \int_a^b x(t)d\mu(t) \textrm{ for a fixed function of bounded variation } \mu(t)\}$. | |
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 23:01, 20 November 2011
linear form, on a vector space L over a field k
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 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
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
[2] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |
Comments
References
[a1] | A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) |
Linear functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_functional&oldid=14314