Linear functional

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 pointwise by 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 a 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