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Difference between revisions of "Linear functional"

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''linear form, on a
 
''linear form, on a
 
[[Vector space|vector space]] over a field k''
 
[[Vector space|vector space]] L over a field k''

Revision as of 15:35, 29 January 2012


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)
How to Cite This Entry:
Linear functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_functional&oldid=19660
This article was adapted from an original article by A.Ya. Khelemskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article